You can find friends in Game Center, and add new friends by clicking the “Friends” tab in Game Center, and then on the “Add Friends” button. Afterwards, you can enter either your friend’s Game Center nickname, or their email adress.
Once you get a couple of friends, you will see more friend recommendations that are based on people in your list of contacts who are on Game Center as well, and people with whom you have a friend in common (similar to how Facebook works). Also, Game Center can give you recommendations based on the games, which you have in common with other people.
During the process of account creation you have the option to upload your contacts to Game Center. This way you will have a number of friends as soon as you create an account and it will help you find more potential friends through suggestions.
Of course, you can also add strangers as friends, but that doesn’t mean they will accept your request. You can do this by visiting leaderboards of certain games, for example, clicking on one of the players and then choosing “Send Friend Request”.
Once you have a new friend, you can view the list of games they played, how many points and achievements they’ve accumulated and how they rank on leaderboards compared to you. If you want to see who’s better in a certain game, you can also challenge them.
Note: There is also an option of removing friends as well. If you decide to do this, simply go to a persons profile, scroll down to the bottom, and click on “Unfriend”.