Getting your PSP "Online"

Getting "online" can mean many things, so I'm going to try and answer them all. 

Getting on to a Wi-Fi network.

To get onto a Wi-Fi network, just scroll all the way left to [Settings] in the home screen (XMB), then to the bottom to [Network Settings]. Then choose Infrastructure Mode, then [New Connection]. If you know the ID of your network, go Manual, but most will want [Scan]. Choose the desired Network out of the list to continue. It may need a security key, which you can enter at a later screen. When you get to the end of the wizard, you press Circle to save the connection, then you can Test it on the next screen. You are connected, and can use that Wi-Fi network for any Internet Functions on the PSP.

Joining PSN:

To join PlayStation Network on the PSP, scroll all the way to the right of the XMB, to the PSN icon, then choose PlayStation Store, it should prompt you for an account, then follow the prompts to create one. You can also head over to playstation.com and somewhere at the top of the Page there should be a Sign In/Sign Up option for PSN. Again, follow the prompts, then enter your newly created login details into your PSP when it asks.

For Online Gaming:

This procedure is very different for different parts of the world. I'm an Australian, so I'll use the Australian method, which is the same for Europe. North America uses a very different method, so you should check your local PlayStation.com.

Australia/Europe:

Unless you are using a very new Game, most games online functions will ask for a "Handle" and "Password". To get one of these, look in the packaging for your PSP for a small code (two parts of 7 digits). Once you have that head over to yourgamingname.com, choose your country, and enter your code on the right where it asks. Then it will give you a Handle and a Password, you can enter into your PSP Game's Multiplayer section to get online.