If you are interested in a research-oriented career, you pretty much have to have a graduate degree. With an MSc degree you might manage a lab or work in an applied field; to direct your own novel research, typically a PhD is required. Most graduate programs allow you to switch from one to the other once you are admitted. Most of the research-focused, high-reputation graduate programs admit mostly or exclusively PhD students. You do not need an MSc degree before applying for a PhD program.
Know that in grad school, unlike college, you usually get a salary and your tuition is covered. You should ask details about how this works when you are interviewing for a program, but you should also seek advice and suggestions on this from your advisor and from every grad student, postdoc, and senior undergrad you can find (especially your lab mates). Do not be shy about this! Your lab is your (second) family, and everyone will be happy to help you or at least recount their own experiences.
Also, look around for other resources. This is your life: research the right information just as you would for a project. There are many websites that give advice on how to get into and do well in grad school: e.g. Matt Might, what grad school is like, a blog by Duncan Watts (arguably a very successful scientist) on serendipity in getting into science. I would also recommend on this website reading the description of what grad school is like.
Once you know you want to do it, figure out where to apply and when. You must start this process early (see the advice on applying); plan to write inquiry emails to prospective advisors in the summer a year before the fall in which you want to start grad school. It will also take you a while to select the programs to apply for - again, seek advice on this from more senior undergraduate students as well as your research advisor and everyone else you know.
Note that in science, and especially for PhDs, I think the most important part is selecting your prospective adviser - no other parameter about the program, the university, the city or the region has more influence on your future career. The reverse is also true: nothing in your application matters as much as convincing your future adviser that you are a great match for her. In programs such as the ones listed below for UA, an Admissions Committee makes the decisions about whether you get admitted; but they will first get input from prospective advisers. Often they will only seriously consider 1-3 candidates per adviser, so you must be in the top #3 for your targeted adviser or your application will not even be read. In addition, advisers will not rank your application and much less put you at the top if they are not 100% sure you are actually applying for their lab. So you must both tell the adviser (in your email, see above) and indicate clearly in your application which adviser you are applying to work with. Indicating two people is usually bad: both will think this means you have only a 50% or less chance of working with them, which means you will not be their top choice. Remember, like everywhere else in life, you both want something from the relationship, and and the adviser may have to choose between a sparrow in the hand (mediocre candidate who definitely wants to work with them) and a pigeon on the roof (superior candidate who may not actually be interested in coming). The smart bet in those cases is typically on the sparrow. ('The sparrow in the hand is better than the pigeon on the roof' is a German proverb, apparently the English version is more like 'A sparrow in the hand is better than two in the bush'...)
Now you are ready to apply. And if you are lucky (and well-prepared) you get interviewed. Follow those links for advice, and read this page to the end: there are several specific, very relevant points below. I can not emphasize this enough: a growth mindset, or a belief that it is hard work, not innate genius, that pays off, is critical. In other words, if you are very smart but you clearly did not even bother to read my webpage to the end, I will not consider you as a graduate student. Similarly, most programs these days expect you to really have not only substantial research experience, but also have thought about a concrete topic that you would like to research, perhaps ideas for several interesting questions to tackle. It is fine to come up with these with help from your prospective adviser in your email conversation before the application; but not being able to mention any possible thesis project during interviews is not a good thing, regardless of your previous grades. (Note that noone expects you to not change your mind after you learn more; but not having any ideas is seen as 'not ready'.)
Also, apply for the NSF GRFP. These are grants from NSF to pay you in grad school, something you don't need to live (because your department/program will come up with funding/a job), but they will make your life a lot easier (because you will have to teach less), AND they will make getting accepted into a good program easier. Assume you will write one in your senior year, and another one (or more) once you've already started grad school. Talk to your prospective advisor (!!) in detail about this as soon as you are admitted anywhere and possibly before.
If you are a student applying to my lab for graduate school, note that there are several possible programs you could apply to. Choose at least two; your work in my lab will be largely independent of which program you are in, but your peers, financing, and required courses will not be identical. As I said above however, all these are only marginally important in my opinion.
You can also choose a minor in any of the above programs or in the following:
If I don't respond to your initial inquiry email, wait two weeks and email me again. Make sure your email address is set up to show your full name as sender (not a nickname or partial name). Do not take non-response personally, I get a lot of emails. However, do try to get me to respond and don't give up after one try.
A few more thoughts on applying that are not included in the general application advice page linked above:
What matters. A good, well-known previous adviser, program, or university matter, but less than you think: your research record and recommendation letters are more important. I told you what I think matters here. Things that matter so little that we are vaguely put off by even seeing them on your CV are which languages you speak; which languages you can program in; which MS Office software you've used; your hobbies.
You must get feedback. No writing is ever good before getting feedback from someone else. That also goes for your CV and personal statement: show it at least to your research adviser, or another professor! Also show it to senior students or graduate students or postdocs - that's one of the benefits of doing a research experience in a lab that has such people.
Recommendation letters. Your recommendation letters ideally come from research-active professors who know you well. Getting such people is one of your goals in college. 'Know you well' means the person has worked with you, ideally over multiple semesters. Class teachers are not good. It is better to get only the required number of letters but have stellar ones than get an additional mediocre letter (not all your reviewers will read all the letters). If you are at all in doubt about your letter-writer's experience in writing recommendation letters, or if you are getting a letter from someone who is not a scientist at a university in the USA with a graduate program, refer them to this page. Remember to take all the work out of writing letters for your adviser - read this.
See the world from your customer's point of view. Inform yourself a bit about how the actual admissions process works. For this, you need to ask someone who knows this answer. Mostly, this is a professor - most likely your research adviser. People who have gone through the process (e.g. graduate students) don't necessarily know why they were admitted, or what the process actually was. This article explains some of it, but note that there are significant differences between disciplines. Don't be shy about emailing your prospective adviser: you are on the same team.
Asking questions is good. Every year, I am amazed that some of my students are not aware of fellowships that their lab mates are applying for, or that I have to repeat the same advice about applying for grad school to three different lab members. Perhaps you don't want to expose your ignorance by asking questions, but this is really the wrong attitude to have when you are working in a lab! You should see your lab mates (both peers and more senior members of the lab) as your collaborators, and everyone is encouraged to help mentor others. In particular, if you are planning to apply for anything, ask around! Have your packet read by others, and read other examples of applications for that program! Take this advice extremely seriously.
A good GPA and GRE scores do matter. Simply because a good GPA shows that you consistently can solve problems and work hard (or as hard as you need to). A good GRE score shows that you compare favorably to your peers from other institutions, banishing the suspicion that a good GPA is the result of grade inflation or a weak prior institution. However, it is not the details of either of these scores that matter. Once you are invited to interview, their importance really drops. If you have some mediocre or outright bad grades, you might mention any events that kept you from doing you work (health issues, family problems: you don't have to be very specific, just mention in which semester(s) these prevented you from achieving the best grades). Or you can say that you realized that this area was not your strength, or be honest and say that your motivation/focus/work ethic flagged: but in both of these cases, you should then pivot to arguments and evidence that demonstrate that you now do have renewed focus/motivation/or that what you are applying for is in fact one of your strengths. Do not say that 'my math grades flagged because I lost interest in math' and leave it at that: this only proves you are intellectually weak and dishonest with yourself to boot. Even worse would be blaming it on the teacher. What I encounter most frequently are mediocre grades early in College that improved once the student found their focus - this is really not that big of a problem and you should address it head-on and honestly in your statement. You might also ask your letter-writer to comment on it in their letter (to confirm that you have changed).
Personal statement: Remember that this is always a bit of a misnomer - admissions committees don't care about you as a person (why would you as a person be better than someone else?). What this should contain is (a) a clear justification for why you are interested in the program you are applying for (use as many different reasons as possible, from research strengths, people you plan to interact with, natural environment, teaching focus, facilities, mentoring program, whatever), and (b) a clear, question-centered description of your past research experience and ideally future research plans. I.e. mention hypotheses tested and broad relevance of your work. Definitely list any manuscripts you will be an author on, even if they are 'in prep'. Make sure you ask your research mentor about whether there are any, and what the title and author list will be. Make sure you mention them both on the CV and in the statement, and make sure you can intelligently comment on the insights of the entire paper even if you only contributed a tiny part to it. This is also true if you are not a coauthor but helped in some way.
Do not focus on particular techniques you have mastered (at least in EEB-type programs, people typically don't care about this at all). Show that you intellectually 'owned' the research you participated in, even if you were just the dishwasher. The only really personal information you might add is an example story of significant personal challenges you have overcome as evidence that you can write well and are a mature individual. You might also mention if you are a first-generation College student or have any other underrepresented minority status.
CV: Academics divide up the world into research, teaching, and service (which includes any kind of contribution to your community as well as science outreach, i.e. communication with the public). Make sure your CV contains any research experiences you have had (whether paid or not is not really relevant), and any presentations or publications you did of this. If you have done any teaching, mentoring, or science communication, list those activities. Then list everything that is contribution to the community, from holding offices in any clubs or societies to charity work/volunteering. Scientists in my experience don't care what languages you speak, which computer software you can use, or which specific lab techniques you have learned (the reason they don't care is that they expect you to learn everything you need 'on the job', each lab's methods are often unique, and PhD research often requires developing new techniques). They certainly do not care about your hobbies. Do list any awards of any kind that you have received, as well as fellowships or stipends.
Other fields. The application form or website might have specific other fields, just as 'Previous research' or 'Research interests'. While you should have all the important information on these kinds of topics in your statement, never leave a field like that empty. Make use of all the space you have to add additional details, particularly where your research is concerned.
Be specific. As I've said elsewhere, choosing a project or research direction is the part about research that requires the most experience, and so you may feel unprepared to do it now. However: in EEB-type programs, a clearly specified research topic of interest is considered one of the things that 'makes or breaks' an application. So, you are not doing yourself any favors by being 'open to different research directions' or listing several broad fields as your interests. Instead, give one general field ('I'm interested in behavioral ecology', say) and then be very specific ('I want to focus on studying cognition in insects', or whatever). You can change your mind later, but for the purpose of the application, demonstrate detailed, well-formed, specific interests and knowledge in that area (by using the correct terms for the field).
People matter. If you worked in someone's lab, say their name. You are talking about this person's intellectual property when you are talking about 'your' research; anyone reading the application will also assume that this person mentored you in a variety of other ways and that you owe them a lot. If you do not recognize this or don't mention their name ('I worked in a molecular biology lab' rather than saying 'I worked with Dr. XXX on YYY'), you come across as ungrateful as well as naive and unprepared for a professional environment. If you worked with additional people (e.g. a graduate student or postdoc), name them as well, but still name the PI (principal investigator, i.e. usually a professor) of the lab you were in. Also remember that scientists use the term 'lab' to refer to the intellectual/social unit, not the room: if you are doing fieldwork, you are still working in the 'lab' of a PI.
Equally, if you are applying to a specific lab/adviser, say their name, give reasons for it, and mention whether you have met that adviser yet or even worked with them. Praise their work and mentoring qualities along with those of the program you are applying to. If you appear lukewarm about wanting to go where you are applying to, this will instantly sink your application. As I've said before, choosing (or at least investigating) potential advisors is one of the most important parts of applying to graduate school - if it appears that you have not done this (e.g. because it is not clear that there is someone who would be an appropriate adviser given your stated interests), this is a massive negative on your application. This is also true for other types of applications, such as UBRP, although to a lesser degree.