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.
The amazing Dropbox can sync just about everything, so why not your music? Well, if you use an iTunes-like app it will cause conflicts with your library files. Basically, you can't have a copy of iTunes running on your laptop when you open it up on your desktop. Fortunately, we've devised a way around this and have an entire guide explaining the process. It's not perfect and requires a little light scripting to help you avoid those conflicts, but it will work. If you don't use iTunes or another library file-creating music player, you can just dump everything in Dropbox without much concern. It can handle files and folders pretty much better than anything else.
The major downside to using Dropbox is the cost. If you can maintain a music collection under 100GB, you'll pay $100 year or $10 per month. While that's not a huge cost commitment, it's a lot to just sync your music. If you have an intensely gigantic collection, it'll cost you more. If you just want to sync music, you may prefer Google Drive. At half the cost, you can store a lot more for less. That said, Dropbox has a lot of awesome features in addition to sync and if you use it for that you might not want to switch. Either way, you can make it happen at a cost.
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.
Sometimes you need a specialized app to get the job done. For those trying to manage an iTunes library, SuperSync ($23, Windows and OS X) and iTunes Sync (Free Windows) can get the job done, but SuperSync is the only cross-platform option and it costs money. For those with folders, any syncing app will get the job done. Crashplan offers free backup and sync software for local stuff. It's certainly not the only thing you can use but one we really like.
If you wanted to sync all your music manually, you could just copy it from one computer to another every time you got a new song. Of course, the point here is to avoid that kind of tedium and find an easier way. If you want to avoid any special software or services, however, you can use watch folders to automatically add music to your libraries with little effort.
Pretty much every music player has this feature, but it can work in a couple of ways. The best way is if you can specify a location where the player watches for new music to add. This way you can set that location and every source can draw from it (over the network or locally) presuming the original file is copied rather than moved.
The other way watch folders can work is if a specific folder is designated by the music player. Any music added to that specific folder will import. (This is how iTunes does it.) That makes creating a syncing solution a bit harder because you can't just download songs and expect them to be added automatically. You can, however, use an app like Belvedere or Hazel to copy newly downloaded music files to multiple watch folders for you to create a sync-like experience.
While not ideal, you can house your music library in a single location (such as a media server) and mount that server whenever you need to open your music library. This doesn't really involve syncing anything, but does allow you to share your music with multiple machines pretty easily. Unfortunately, you still can't have two copies of library file-creating software (you know, like iTunes, the usual culprit) open and you'll have some trouble getting to your music on the go (unless you have a fast upstream connection for your server). Nevertheless, if you just want easy local access and can sync your tunes to your mobile device for listening on the go, a networked option may suit you just fine.