There are a lot of ways to get your music collection on other computers but none of them are perfect. Some will work better than others, but that'll depend on how you store your songs. If you just keep a collection in folders, you have much more flexibility. If you use a jukebox app like iTunes and it creates library files to keep track of everything, you'll cause yourself some trouble. You also have alternatives that aren't quite sync but at least give you access to your music. Let's go over your best options and you can decide what suits you.
Whatever you choose, it all comes down to finding the sacrifices you're willing to make. Syncing music is no simple task, and nobody's really found a masterful solution to the problem. Nevertheless, you can make it happen if you're willing to do some work and understand you can't have everything you want.
-Dropbox (and Other Syncing Services)
-iTunes Match (and Other Cloud Music Services)
-Use an App
-Use Watch Folders
-Stick It on a Network Drive
-Stick It on a Local Drive