Read what I wrote about grad school; most of it still applies now.
In some ways, being a postdoc is easier than grad school: you have fewer conflicting priorities work-wise (typically no classes to take or teach), and the transition from grad school to postdoc is a much smaller step than from undergrad to grad. However, often postdoc postions are not very long, so you basically start looking for the next thing as soon as you start, and this uncertainty can be stressful. In addition, often you'll have more personal commitments now, perhaps even a family, which makes you less flexible both in the short and long term. Also, connecting to others will be harder, because you are not automatically getting together in a classroom. BUT, by and large, this should be a time when you can really concentrate on research, and are relatively independent (and skilled enough that this is a benefit) in choosing what to work on.
A good postdoc advisor will encourage you to find your own research niche (one of the most important outcomes of a postdoc period) and allow you the freedom (and give you at least some resources) to explore it. However, you should still seek a lot of guidance on what questions to pursue, on university politics, etc., and you should still improve at the same skills listed as learning outcomes above for grad school.
Efficiency: If you have not already done this during grad school, you must really learn to be productive and efficient without external pressure. Check the page on applications to get an idea of what outcomes will be expected from you. Search committees know that the amount of additional responsibilities when going from a postdoc to an independent faculty position is high, so you must really prove that you are highly productive, able to multi-task (e.g. teach and do research during the semester), able to delegate (advise students and get them to be productive), and able to prioritize.
Connect to other postdocs: be proactive. Start a mailing list, happy hour, or other social events. Do not look down on graduate students (or undergrads): they are only a few short years away from being your colleagues.
There are many other blogs and websites giving advice, e.g. this one:
Bullet-point advice for how to get a job and tenure in academia (by John Boothroyd).