Terminology:
PSP-1000 = Phat, old, heavy PSP. Comes with firmware varying. Hackable.
PSP-2000 = Slim, newer, light PSP. Comes with firmware 3.x.
PSP-2000 TA088v3 = Newest PSP-2000 model. Comes with firmware 4.x. Unhackable.
PSP-3000 = Slim, newest, unhackable PSP. Comes with fimmware 4.x.
Well, first, get a pandora (tool) battery, if your firmware is over 3.50. If your PSP came with 4.x firmware, you are screwed, don't even read this guide. But, if your firmware is under 3.50, you can downgrade without a pandorah. But, pandorahs have a distinct advantage: You can unbrick (unbreak) your PSP any time you want if you fuck it up.
Dealextreme sells excellent stuff according to forumers
DealExtreme: $11.71 Hybrid 2-in-1 Digitally Activated Pandora PSP 2000 Unbricker Service + Normal 1200mAh Battery
This is a digitally activated Pandora, restore it with square in the pandora menu or something (I forgot, do the option where it says "Restore pandora to original battery" or "Restore IDstorage keys" or whatever.) You do not need TimeMachine for this one. $11.71. PSP-2000 only - Not compatible with PSP-1000.
DealExtreme: $9.50 PSP Unbricker Pandora Battery for PSP 1000/2000 (1800mAh/3.6V)
This is a non-digitally activated battery. You will need TimeMachine to boot this one - it's just one extra step anyways, not that hard

there are tutorials for it everywhere!
$9.50. PSP-1000/2000. Fits with the battery door for PSP-1000. Does not fit with battery door for PSP-2000.
DealExtreme: $17.33 Pandora PSP Unbricker Service Battery + Memory Stick Duo 2GB Combo
This is the above option with a 2GB memory card
$17.33. PSP-1000/2000. Fits with the battery door on for PSP-1000.Does not fit with battery door for PSP-2000.
DealExtreme: $22.28 Pandora PSP Unbricker Service Battery + Memory Stick Duo 4GB Combo
This is the 2nd option with a 4GB memory card
$22.28. PSP-1000/2000. Fits with the battery door on for PSP-1000. Does not fit with battery door for PSP-2000.
DealExtreme: $10.50 PSP Unbricker Pandora Service-Mode Battery for PSP Slim/2000 (800mAh)
The PSP SLIM ONLY battery. Fits with the battery door on for PSP-2000. Not compatible with PSP-1000.
DealExtreme: $11.96 Pandora Service Mode Unbricker Battery for PSP 1000/Phat (1800mAh)
The PSP Phat, has the same battery capacity as the orignal sony batteries - install TimeMachine and you'll be good to go with two almost-identical batteries. PSP-1000/2000. Will not fit with battery door for PSP-2000. Will fit with battery door for PSP-1000.
Then, get a MMS (magic memory stick)
» Rain’s UltraLite MMS Maker v1.1 Released PSP Hacks · PSP Homebrew, PSP Downloads, PSP Mods
Then, downgrade! n.n
Choose the "4.01" option.
After that, do a network update (it's patched, so it does hacked firmwares only! It's okay!), and upgrade to 5.01 M33-3.
Install TimeMachine 0.1 if you want to - that makes your pandora battery usable as a normal battery, as well as offering you protection from insta-bricks by allowing you to boot into pandora or another CFW instantly. Guide here:
Connection Problems
Download here:
» TimeMachine v0.1 Released PSP Hacks · PSP Homebrew, PSP Downloads, PSP Mods
As I wrote, the PSP-1000/2000 combo batteries are the PHAT BATTERY TYPE and WILL NOT FIT IN YOUR SLIM PSP WITH THE BATTERY DOOR. The PSP-2000 only batteries will not work with the Phat due to space issues.
After doing this, format your PSP. Just save the files you want and format your PSP. It'll show up with new folders, crucial if you want to run homebrew and don't want to reasearch all the folders you need to make.
If your PSP is a PSP-1000 (phat), Install the 1.50 kernel addon for the newest firmware. 5.00m33 kernel addon here:
File Details-1.50-kernel-add-on-for-CFW-5.00-PSP -
Also, check out
PSP Hacks · PSP Homebrew, PSP Downloads, PSP Mods for new updates on homebrew and such.
And customizepsp.co.uk for custom themes (Current version of cxmb is 3.3, it works with 5.01m33)
Oh, and if you have a Slim, get the Tool battery for the slim. Bewarned there are unhackable PSP's. What model is your PSP? If it's PSP-3000 or PSP-2000 TA088-V3 we have a problem. But if it came with less than 3.x you are okay.
PM me for my email/skype for further info.
Another copy-pasted answer by fishyfish