Get a PSP model under 6.35. It HAS TO BE 6.35 or lower to hack fully.
Get an old non TA-88v2 motherboard 2000 or 1000 model will give you the most features, making original 2000s the most sought after (and in my opinion, best PSPs).
a TA-88v2 (or v3) and higher revision Motherboard is one that came in the PSP 2000s starting with the Blue Madden PSP bundle and Kratos Red God Of War bundle (NOT the new 3000 black and red one, an older one).
The first run was fully hackable, but the second run and so forth were not. Until Half Byte Loader, Total_Noob, Wololo, Viruous Flame and ColdBird came on the scene.
They made it so TA-88v2and+ can now be "fully hacked" essentially (I know, not all of them at once or all together I am just saying they contributed A LOT). They can run what is now the modern Custom Firmwares 6.20 PRO-5 and 6.35 PRO-5.
6.20 can be run on any non 9g (Ninth generation) mobo, you can use the program "PSPident" to check your PSP's Mobo. If it is not 9g, you can run 6.20 Permanent patch, which will give you the first "Full Custom Firmware" for ALL PSP MODELS (besides 9g).
The old 1000s and 2000s can use that, or TN-D if you just want Homebrew + PSX support (PRO-5 has great but not perfect PSX support as well). They can also use the old school Full Custom Firmwares known as 5.00 M33 (or Prometheus variants to play newer games), 5.50 GEN-D4 (or Prometheus variants to play newer games). There are plenty more to name but those two are all you need to know.
If you use 5.00 M33, there are tons of old school plugins and PSX games you can use via Popsloader (a mod for PSX support, choosing the right FW's Pops to load almost made perfect compatibility). Same goes for 5.50 GEN.
Note to get to M33 or GEN you must use Hellcat's Recovery Flasher to jump down to 5.00 or 5.50. ONLY 1000s and NON TA-88v2 and up MODELS CAN BE FLASHED WITH THOSE FWS.
for 3000 and Go models it's simple. you update to 6.20 (or downgrade, IF YOU CAN using Davee's PSP Downgrader) and run 6.20 PRO-5 installer. then run perma patch, easy, you get BEST PSP game support and good PSX support. Only 2 of 30 games I tried would not work.
If you can't run perma patch or 6.20 you can use 6.35 PRO-5, which is the same FW but you must use what VF and Coldbird call "Fast Recoevery" to load the CFW upon each startup of the PSP after fully shutting down.
If you need any help or clarification let me know.
EDIT: History lesson aside and simple answer is: ANY PSP under or on 5.25 is hackable fully, just run PRO-5 on it and done. Run 6.20 version on 6.20 and perma patch it for a permanent CFW, done.