The first and main one is don't drop it...every single hit is bringing it closer to death. Especially if your mp3 has a hard drive in, like the ipod classic 160GB...one drop=99% dead on the spot.
Secondly, when the dock port where you plug it in to the computer or to charge it, always take it straight out without bending, because it damages it severely if you bend it and take it out the same time.
Careful with the screen, don't put it on hard surfaces or it will get scratched up.
And lastly, every mp3 player just like every other piece of electronics or mechanical piece of equipment has an X amount of times thats its designed to work before it fails. If you press the buttons on the mp3 player to often, they will fail quicker, if you plug it into the dock too much, it will fail quicker, if you turn it off and on to often, it will fail quicker. It might still last long, but not as long as it would otherwise.
And lastly try to keep it in reasonable environments. Too much heat can damage it. If its 95degree day outside and you park your car out of a store under the sun, you car will heat up to 120+ degrees in minutes...don't leave it in your car, bring it with you. Especially the battery will get damaged under too high heat. Don't keep it in too cold environments either...room and outdoor temperatures only. Too much humidity is also bad.
if you follow those steps, your mp3 player will last 50years...so you don't have to follow all of them, just try and it will last long enough until you decide you're done with it and want a new one.