Zotero offers free database synchronization across devices, but if you want to synchronize more than 300Mb of attached files through Zotero’s official storage service, you will have to subscribe to a paid plans.
For not searching enough in Zotero’s documentation, I initially thought Zotero’s official storage service was the only way to go. For a while, I had been using a hand-made iffy solution relying on a dropbox-like synchronization of my “Literature” folder, in order to have my attached files synchronized without paying. I discovered the hard way that this “solution” was not without problems (see https://forums.zotero.org/discussion/56023?page=1#Item_4).
Then, as I was working out the problems with my hand-made solution, I rediscovered that Zotero’s folks are really all you can expect from open-source developers. Among other things — and unlike folks at Mendeley, they do not want to trap you into using their own storage service. This means you can safely use a free third party storage to synchronize your attached files. Just follow the procedure described on https://www.zotero.org/support/sync, under the title WebDAV.
Zotero even maintains a list of third party storage providers which are known to work fine with Zotero : https://www.zotero.org/support/kb/webdav_services.
I personally chose 4shared and have been happy with it so far (4shared’s connection to Zotero used to be a little hard to set up — see http://www.steletch.org/spip.php?article83, but this problem was solved in later versions of Zotero).