The PS Vita has an app--pre-installed when you buy the unit--that makes keeping track of your PlayStation Network friends simple. If you've already got a PSN account and a list of friends, that list will be available when you connect your PS Vita with your PSN account. You'll be able to make new friends from the app, check their status and view their activity, make comments, and communicate via other PS Vita apps connected to the Friends app.
The first thing you'll need to use the Friends app is a PlayStation Network account. It's free to sign up for a basic account, and if you're going to want to download games or demos, you'll want one anyway. The start the Friends app, tap its icon on your PS Vita home screen to open the app's LiveArea. Tap the big rectangle in the middle that says "Start" to launch the app.
Once you've got a PSn account, all you need to add a friend is the person's PSN ID name. When they give you that, go to your friends list (you can open this even if you don't have any friends added--you'll just see your own ID at the top) and tap the "options" button--it's the one in the bottom right corner with ". . ." in white on black--and select "Send Friend Request". Enter the person's ID (if you need help with the on-screen keyboard, see this article) and tap "send". This will send a request. You'll now see the person listed in your friends list, at the bottom with any other pending requests. If they accept your request, their name will be moved up to the active friends, and if they decline the request, they'll disappear from your list.
Requests from others will appear on your friends list, at the top. Tap "Friend Requests" to look at the them. To accept the request, simply tap the "Accept" button next to the person's name. To decline the request, you'll have to tap the person's ID name to open another screen, where you'll see more information about the person. To refuse the request, tap "Delete Request." You can also accept from this screen, so if you're not sure if you know the person, tap their name for info, then decide.
This is also where you block other people you don't want to interact with. On the person's info screen, tap the "options" button (bottom right corner with ". . . " on it), and select "Block." If you've blocked someone and decide to unblock them, do the same, but select "Unblock" instead.
Several other PS Vita apps can use your Friends list, and even let you send friends requests. Party, Near, Group Messaging, Trophies, and even some games let you interact and communicate with people on your friends list in various ways. If you access the PlayStation Network from a PS3 as well, your friends lists will synch, so even if your friends only play on PS3, you can still see what they're up to and even send them messages. To check on your friends, start the app, and look at your friends list. Tap the person's ID to view their profile.
From a person's profile, you'll be able to see their avatar and whether or not they're online (a blue dot means online, a red dot with an x is offline, and a yellow dot with white clock hands is "unavailable"). You can also see is they're currently playing a game (a PS3 controller icon shows if they're playing a PS3 game, and a PS Vita icon shows if they're playing a PS Vita game). Some games have thumbnails that will appear when someone is playing them, too.
Depending on the person's security settings you may or may not be able to view their friends list. In the top right corner, you'll see an icon that looks like a speech balloon. Tapping that will launch the Group Messaging app and let you send a message to the person. You'll be able to see their latest activity, check out their trophies, launch the Near app to see how close thy are, and see your shared play history.