Parts
Depending on the package you have purchased, each PSP will come with different things packaged inside. Unpack all of the components before you begin setting up. At the very least, the PSP itself should be in the box with a power cable, a battery and the manuals. If you have purchased a bundle of some kind, it may also include a Memory Stick Duo memory card, one or two UMDs, containing either a game or videos, or some other accessories. Set aside the UMDs until you've got the console working.
Assembly
There should be a small button on the side of the PSP which will open the battery hatch. Flip the console over and open the battery compartment, and insert the battery into that slot. There's a knob on the side of the battery that should fit into a corresponding slot in the compartment--insert that first. Next, if you have a Memory Stick Duo card, locate the hatch on the other side of the console and slide it open. Insert the card, press until it clicks, and close the hatch up. The PSP is now assembled, but it still needs power. The power cord comes in two pieces, so connect them before you start charging the PSP.
Initial Setup
After a few minutes' charge, power on the console. There's a small switch on the side which you can slide up for the power. The screen will flicker to life, then display a PSP banner. After that, the initial setup will ask you to come up with a nickname for your console, input the time, date and time zone, and select some other initial settings--such as your birthday or parental controls. Follow these on-screen prompts to make the console your own. If you want to change these settings at a later date, you can find them under the "Settings" section of the home menu bar.
Home Menu
When you're finished with the initial setup, you'll be brought to the XMB, or XCross Media Bar, as Sony calls it. This is effectively the home menu for the PSP console. On it, you'll find a number of features: Games, Movies, Photos, Music, Network, the PlayStatyion Network, Extras and Settings. "Games" is where you'll find any games you download to the console, or the UMD game disc when you insert it. You can also download music, photos and videos from the PlayStation Store, or upload them from your computer with a USB cable. "Extras" is where Sony adds new features, like the comic reader, to the console. "Network" houses your web browser, Skype (for the PSP 2000 and 3000 models) and other web features, while the PlayStation Network connects you to Sony's store and your online gaming account.
Internet Setup
Under the "Settings" menu, which also contains many other personalization and software settings, you'll locate the "Network Settings" option, which allows you to set up the connection to the web. Note that using the Internet will use up more battery power, which is why the WLAN switch on the console was added. You can switch it "On" when you want to be online, and "Off" to save some extra battery power. Flip that switch and head to "Network Settings." Set up a "New connection" and then select your favorite Wi-Fi network to connect to. Follow the on-screen prompts to select and input any security settings for the network. Finally, test the connection. When it works, your console can connect to the web from this network at any time.