I quit social networking sites because they made my life significantly worse. If you really need to use them, you can, but don't worry. There is a wide variety of ways to use them, ranging from adding hundreds of people to just a few friends.

So should I just create an account and add every single person I am even tangentially acquainted with?

Yes, you can do this, but you don't have to. This is one reasonable way of using the site that a lot of people use, but it's also common to restrict things to people you know better.

Is there a feature on facebook where you can hide who your friends are?

YES. Absolutely. And it's an essential feature. If you do use Facebook please pay close attention to the privacy settings. You can make everything about yourself private, to the point where no one else, even your friends, can see anything except messages you specifically send them.

Is it appropriate to ask someone you just met to friend you?

Yes, it's pretty common to do this, though you may be surprised by how many people don't like to use these sites.