Several issues here.
Yes, you need some sort of an amplifier to drive speakers. A receiver is the best bang for the buck.
These come in plain vanilla two-channel stereo as qell as the full-blown HT scenario. For MP3's, all you really need is two channels. Decent Two-channel stereo receivers NIB start at < $100 (best buy's Insignia brand, some Sherwoods and maybe others).
Likewise, you might find a bargain at a garage sale o or some other means. With these, all you need is a cable from a dollar store to connect an MP3 player.
As for sound quality, well, it'll be light years better with a receiver and speakers than it is through your headphones or computer speakers. Will YOUR* MP3 be the end-all and be-all for sound? Maybe, maybe not, but at least you'll now be set for other sources, the receiver's radio, a CD player, your computer, perhaps a turntable, and who knows what else.
* Depending on the compression rate you used to jam the music onto your MP3, you can get some pretty decent sound from it. The lower the bit-rate the "less pleasing" your sound is likely to be. I use WMA on a Sandisk Sansa and it rivals a CD player.
