The most reliable way to move your entire library to another computer is to copy the Zotero data folder from your first computer to your new computer.

To locate your Zotero data, open the Zotero preferences, go to Advanced → Files and Folders, and click “Show Data Directory”. See here for the default locations of the data folder.

Be sure to close Zotero on both machines before copying the Zotero files. If you've already opened Zotero on the new computer, there will already be a Zotero data directory with an empty database, and you should delete the whole data directory before copying the new directory to the same location.

You can also sync your library between multiple computers via the Zotero servers. Make sure you use the same username on all computers.

If syncing files, you'll need enough online storage space to fit all files in your library. Zotero will warn you if you hit your quota, in which case you may need to delete some files, add a storage plan, or transfer your library using Option A above.

Exporting and importing your library (for instance via Zotero RDF) is not a recommended option. None of the available export formats allow for a complete transfer of your library data, reimporting will break connections with any existing word processor documents, and if you use syncing later you will end up with duplicates of any imported items.