If you've been away for a while, maybe on a desert island
The one thing you need is a lot of storage. Figure about 200 songs per Gig, and then figure how much music you have. Keep in mind that once you get into the habit of listening to your MP3 player, you'll start collecting a lot more songs. So get at least four or five gigs, so you can save plenty of music on there.
Next thing you'll need to be worried about is how long the battery lasts. This is actually two different concerns. One is how long the battery lasts on a single charge. It should be at least 8 hours. That way, you'll be to listen to it throughout the day without worrying about it conking out. The other thing is how long the battery will last before it no longer can hold a charge. For most batteries, this is a year or two, depending on how much you use it.
When you use your MP3 player will have a huge impact on what kind you should get. For example, if you are primarily going to be listening in the gym while working out, or jogging, then you'll want a small one you can strap on your upper arm. On the other hand, if you're going to be listening to studying, then you can get a bigger one to lay on the table.
One thing that's definitely something you need to look into is how easy it is to add music. Ideally, you'd want to just plug it in to your computer, and then simply drag and drop from one file to another. This is the easiest way. Beware of MP3 players who need their own special software that takes forever to load.
These are the key things you'll need to be concerned about when shopping for an MP3 player. Just take your time, and before long, you'll have your headphones on and will be dancing down the street.