I took some quick pics to keep track of what I'm doing. First I started with my non TA-082 vanilla PSP. My PSP was shipped with version 1.60 if I remember correctly. From there I updated a few times for different games, and ended up with a firmware version of 2.71.
Then I used the "2.71 Downdater" by Dark_AleX/Mathieulh/Yoshi. Which uses a know TIFF overflow exploit to revert the firmware back to 1.50. 1.50 can be easily tricked in to running unsigned code in unprotected mode, making it the perfect jumping off point for installing a new and fancy hacked firmware. This is by far the scariest step, since it involves using an overflow to over write flash memory (which if corrupted would brick the system). It takes a random number of tries to get the TIFF exploit to work, sometimes it just freezes up. The only way you "know" the downdater is working is when the screen of your PSP goes all red and stop responding... and then it crashes which means it worked.
1.50 Firmware PSP (soon to be hacked)
After the downdater finished. I had a fancy new 1.50 firmware PSP, so I went on to upgrading it to "3.40 Open Edition Revision A" also by Dark_AleX. 3.40 OE-A is very cool, Dark_AleX basically made a system that combines the easily hackable-ness of 1.50, access to all current games, and you get all the fancy features of 3.40. So it's the best of both worlds.
So now I'm in the fancy world of Homebrew PSP. The first thing I installed was SNES9X, SNES games are mostly playable. The next thing I tried was FileAssistant++, man it's nice to have a function file manager for my PSP. Now I'm just looking for a better media manager.
UPDATE: I just installed Installotron, which installs homebrew right off the web using only the PSP. It's hella cool :)