You can downgrade your PSP to 1.50 using either Pandora or an unpatched copy of Lumines. After you've downgraded to 1.50, you can then install custom firmware. From there, you can either keep using custom firmware or install the official 3.71 firmware. Okay, that was a bit oversimplified but when you've got the necessary tools, then just inquire again later.
Using a modified version of Pandora (Dark_AleX's Despertar Cementerio v3), it is also possible to install official 3.71 firmware or custom firmware 3.71 M33-2 directly. I believe this would be the best and easiest route provided you can get your hands on or create the battery. I can supply you with the files for both regular Pandora or Despertar Cementerio should you need them. Just PM me or something.
I'd suggest you read the following guides:
Basically, if you only need to play UMD games you buy and do Sony-sanctioned things on your PSP, you can just go with official firmware and save yourself further research in the future.
If homebrew has whetted your appetite, then please, do read the Guides' section and come back to post some questions if you need further help or clarification.