What makes music sound better? Who cares?

Speakers. If you want it to sound better than a boombox you can get separate speakers. Sometimes there are good ones for cheap at thrift stores (Salvation Army, Goodwill, etc) for about $20.
Amplifier. Separate speakers need a separate amplifier. This little one (the Lepai 2020A) is loud and sounds very good and is about $25. You could also find an older one used for cheap.      
Subwoofer. An "amplified" or "active" subwoofer can plug into just about any sound system. You plug the amplifier into the subwoofer, then the speakers into the subwoofer. For about $200 you can get this thing, the Dayton 1500, which people say is the best value available.
Bluetooth. Normally amplifiers don't have bluetooth. You can add HD bluetooth (APT-X) to an amplifier for just over $20 and play music from your phone.  The Avantree works fine and listed on amazon.
Total. Is about $270, or just $70 without the subwoofer. All these things last about 10 years or more.