If you'd like to browse (somewhat at least) the internet on your PSP, this method from 2009 still works as of August 2020. These days, this is pretty much the only way to access any page, since the built-in NetFront browser won't load pages with newer SSL certificates (and since most websites are HTTPS-enabled now, that means most things won't load anymore).

This guide is based on my old one posted here.
Please note: I've never this on the PSP go (N-1000) so I'm not sure if it would work with it.
All browser activity in Opera Mini goes through their servers, so I wouldn't recommend anyone to use it to submit personally identifiable information. This would include logging in to websites such as Facebook, your bank (although it would be impressive if you managed to, honestly), or similar things.

Supplies

PSP with custom firmware. Custom firmware's required to run PSPKVM.
Computer. To read this guide and to send a few files to your PSP.

USB cable compatible with your PSP. If you have a PSP go, you'll need the proprietary cable that comes with it.
Memory Stick (Duo or PRO Duo, not needed if you have a PSP go though you could probably copy it to a Memory Stick micro if you have one). You'll need about 40 MB free, although I (still) haven't really checked just how much space PSPKVM and Opera Mini take after setting them up.
Internet connection, both on your PSP and your computer. Be careful, the PSP will only connect to a 802.11b access point secured with WEP or WPA (either TKIP or AES). In this age of 802.11ax and other more exotic wireless standards, the presence of 802.11b in your router might be rare depending where you live, so keep in mind that you may not be able to connect if you have modern wireless equipment. Same with wireless security standards. If your access point only has WPA2 or newer, you will not be able to connect.

Step 1: Download the Required Files

You will need PSPKVM. The last version can be found here. Please download the file named "pspkvm-bin-0.5.5-fr-cfw-allinone.zip" (unless you're upgrading from a previous version, in that case get the "pspkvm-bin-0.5.5-fr-cfw-upgrade.zip" one instead).

Opera Mini can be downloaded here. Select "Opera Mini 4.5" on the "Choose an application" dropbox, and select your preferred language. Personally, I haven't tried any other than English, but unless it requires a vastly different script (such as CJK, Arabic or similar) I'd imagine it would work. Download both .jar and .jad files.

Step 2: Transfer the Required Files

Connect your PSP to your computer and extract the PSPKVM zip you downloaded to your PSP's /PSP/GAME/ folder. You should then have a pspkvm folder, with subfolders appdb, inst and lib inside it.
Copy the previously downloaded Opera Mini .jad and .jar files to your PSP's Memory Stick (or internal storage, if using a PSP go) root (where you can see all the other folders).

Step 3: Setup PSPKVM

Disconnect your PSP from your computer, and from the GAME menu, launch PSPKVM.
Please note: by default, PSPKVM uses O as the confirm key, not X. You can change this behavior later.
Before doing anything else, go to System Midlets and then choose Network Setup. Connect to a network.

If you want to change the X and O buttons behavior, go to System Midlets > Configure VM. From here you can choose your on-screen keyboard as well.

Step 4: Setup Opera Mini

In PSPKVM, select System Midlets > Find Applications. Select "Install from Memory Stick", and find the "opera-mini-4.5.XXXXX-advanced-en.jar" file you copied earlier. Select that and the installation process will begin.
The buttons shown in the lower left and right of the screen are triggered using SELECT and START respectively.
Choose device J2ME 480*272 when prompted.

Opera Mini will launch, show a splash screen with "Installing..." on it, and then it will ask you to accept the terms and conditions. Once you accept that, you will have successfully installed Opera Mini and it will be listed under Installed Midlets.

Step 5: Finished (but You Can Change Some Things If You'd Like)

You can now use the browser, but you can also change the browser's settings. After launching the browser, go to Menu > Tools > Settings.
There are options related to synchronizing your bookmarks and history with the desktop browser, but those features are no longer functional.

PSPKVM includes the (horrifyingly ugly) DejaVu Serif fonts, and for some reason Opera Mini makes use of them, displaying all web pages and the browser's UI with them. If you want to change them for something less jarring, you can replace them by overwriting the files prop.ttf, prop_b.ttf (bold), prop_ib.ttf (bold italic) and prop_i.ttf (italic) located in the /PSP/GAME/pspkvm folder. This may break other JAR applets you may have installed (or may install), but I haven't tested it. In my screenshots, I replaced them with Noto Sans, which you can get free from Google Fonts. Make sure to rename them accordingly, or you might end up with all-bold fonts or something.

In case you're wondering, you can install Opera Mini 8, but it's so slow that it's just not worth it. It doesn't even have color themes (okay, Opera Mini 4 has just three but still). It also takes a long time to install, so just don't.