Brewology

Alright, so I am fluent in Java but apparently you can't really do anything with that in terms of programming stuff for the PSP. So I've spent my time learning C++ and I think I'm ready to enter the world of the PSP homebrew developers. What I need to know is how I go about writing the code (i.e. IDE and compiler). Basically what I'm asking is how to I go about writing a program to run on the PSP. Also if someone could point me to a site or link that has common syntax used by the PSP system that would help a lot. Thanks for all your help.

You will need some kind of Linux OS or a Linux runtime environment and download the PSP SDK that's online.
I use C when programming for it so I'm not too sure how much the syntax differs but I know that most of the PSP-specific functions are the exact same; So, they're pretty easy to pick up.
I don't think there are many people out there coding in C++, so if you haven't gotten far in learning it, I'd suggest switching over to C instead. You'd get more help from it.
When you download the SDK, you'll get a lot of files telling you what each function is made to do.
Go here for some tutorials: http://www.psp-programming.com/