First of all, congrats on choosing psychiatry - great choice!
Here are a few things I can think of now:
1. Enjoy the process.
One thing to know up front is that it is very likely that you will find it a much friendlier and enjoyable process than applying to medical school (and perhaps also if you compare with the process for some other specialties).
2. If you are interested in a particular program that is not at your current school, I highly recommend doing an away elective there.
Sure, there is always the possibility that you won't get along with the team and won't like it or fear that they won't go to bat for you with the residency director/ad com. However, it is better to know whether it is a good fit on both sides before you apply, and it is also much more likely that your away elective will increase your chances of being ranked at a program.
3. Get a lot of people to read your personal statement, and make sure it reflects your personality.
I found through my own and others experience that psychiatrists take the PS pretty seriously, and use it for a jumping board for the questions they ask you.
4. Interview at a range of programs in places that you want to live.
I think it's always a good idea to have a good range of programs that you apply to and rank. While psychiatry may not be as "competitive" as other specialties, there are certainly programs out there that are highly selective, and you don't want to put all your money on just a few programs.
5. Make your interest in the program known from the start and through the process.
If you do an away elective somewhere and are interested in the program, be sure to meet with the residency director at some point during your elective and express your interest in learning more about the program. And when it comes time to rank, definitely tell your first choice program that they are number one. Psychiatry programs are small enough that one or two places up or down in the rank list can make a big difference in whether you match at a place or not.
Best of luck!