iTunes users can turn to Apple and use iTunes Match to sync all their music for $25 per year. While syncing doesn't precisely describe the service, in that you have to manually designate songs on each machine (and mobile device) if you want a copy, the effect is the same. Your entire collection shows up everywhere and copies are stored in the cloud. If you want a song locally, choose to download it or just play it. iTunes Match offers a simple solution to keeping your music everywhere at a great price. The only downside? You have to use iTunes.
Unfortunately, other streaming services don't do the job as well. Google Music and Amazon Cloud Player simply sync your music to the cloud. You can access that music on virtually any computer or mobile device, but through streaming or tedious downloads (often of a limited number). If you don't mind relying on streaming when you want to play your music on another computer, this might be the option for you. Google offers their service for free while Amazon charges $25 a year like iTunes Match. Both are good services and have handy mobile apps which do allow saving local files, so either choice can provide a good experience.