It wasn't that long ago that I was a new postdoc (shut up! It really wasn't that long ago). IME being a postdoc is awesome! You learned the basics of how to be a scientist as a graduate student. Now is you chance to develop your research skillz in a protected and supported environment. It is about as close to a CareBear Tea Party as you are gonna get in this business.
Being a postdoc is not always easy, and I know there are some disgruntled postdocs out there. There can be issues with your mentor, issues with the science. There are always going to be struggles as you work to find your own path and develop your career. It is hard work, and can be scary. You will most likely move to a new city. Maybe you moved to a new place for grad school, but this time you will not be entering with a bunch of classmates. You will be thrown into a fully-formed lab that may already have political baggage and/or a defined hierarchy. At some point you will start wondering what is expected of you. There is no universal answer, because every lab and every postdoc is different. But, here is a list of advice based on what I think helped me when I was a new postdoc and that I hope postdocs that join my lab will follow:
1. Don't assume that you know more than techs or graduate students because you have a PhD. Coming in as a n00b postdoc with a superiority complex will not help you. You will need help from your new colleagues, so don't act like a douche. Be a good lab citizen, and build a relationship with the lab folks.
2. Be ready to learn something new. Maybe you enter a whole new field and have to learn brand-new techniques and approaches. Perhaps you are in a lab that has some overlap with what you did as a grad student. Learn how folks in your new place do things. Realize that there is more than one way to do most things...and yours is not necessarily the "best".
3. Take the initiative. Don't expect for someone to "give" you a project. Find and read the important papers and come up with ideas of your own. Consider advice from your new PI, and people in the lab, but argue for your own ideas and approaches. Own your project. Find fellowships that you can apply for. Apply for them.
4. Get to know your colleagues. Find other new postdocs and get to know them. Actively build a network. You are going to need mentors, advice, and letter writers. You will have to talk to people that are not in your lab (or your institution) to do this. Don't wait to approach people until you need something. Build a relationship from the beginning. This includes other postdocs and grad students as well as faculty.
5. Think about what you want to do with your career. It is great if you want to stay in academia, but try to imagine a Plan B. If you don't want to stay in academia, figure out what you want to do. Then find out what skills you need to develop and find opportunities to do that. This can be teaching, writing, working with policy, interacting with tech transfer, etc.
6. Be realistic. It takes time to get a new project up and running. Science (and career development) takes time.
7. Don't play it safe. Your postdoc is a great time to try a high-risk/high-reward project. Be creative. You can have a back-up, "safe" project, but don't shy away from trying something "hard". Try to avoid the temptation of taking on the easy, obvious, "can't fail" project.
8. Ask questions. Lots of questions. At lab meetings, in seminars, walking through the hallways.
9. Make sure you know what is expected of you. Many of the poor postdoc-mentor relationships that I have seen stem from miscommunications. You need to make sure that you know if there are expectations about how many hours you are in lab. If you want to stay in academia, make sure you know whether you will get to take your project with you (have this conversation early in your postdoc, before it is clear how awesome your project is. Even if it is non-binding and is not a guarantee that the lab will not compete with you).
10. I honestly think that most PI want to be good mentors. Help us out! Be a good mentee.
Please add to this list in the comments! I have no doubt that there are other nuggets of advice out there for the newly-minted postdocs.