We have PSP section (
http://gbatemp.net/index.php?showforum=44 ) but in general
PSP1000 fine
PSP2000 earlier models are fine- later models are more troublesome
PSP3000 - can be done.
PSP go aka PSP4000 in some circles- no real chance.
There are hardmods but nobody uses them. Everything revolves around a custom firmware.
You have chosen to join at a bit of an odd time:
the earlier king of custom firmwares was M33 but that has now stopped (although it does work for just about every game released up until this summer out of the box and with a bit of fiddling it works with most of the rest)
Today the gen firmware is the must have firmware which should work with every game bar the ones that cause troubles.
If you are before official firmware 5.03 (or at it) then you have the chickHEN exploit.
In the case of PSP1000 and hackable slims (the earlier models of PSP2000) then use chickHEN to get into a homebrew firmware, from that install a custom firmware.
In the case of the later models flashing a custom firmware like the above wins you a brick. Use chickHEN to get into the homebrew firmware (that can not run games) and then use a specialist version of the custom firmware to run it while you have power (including standby mode). Not ideal but it works for lots of people.
If you are above 5.03 on a PSP 3000 or unhackable slim then you are probably out of luck (there was another exploit but it is very unlikely you have the ability to do it (it was patched in an update)
If you have a hackable slim or PSP1000 and are above 5.03 then you have the option of using a pandora battery to load whatever firmware you like including a custom one.
Poke around our PSP section if you want but if not there are loads of PSP blogs (perhaps not anything like gbatemp for the PSP but they do for what you want), a couple from my bookmarks
http://pspcustomfirmware.com/
http://pspslimhacks.com