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. Once you've found the Zotero data folder, save a copy of it to a Google Drive folder, a USB or an external hard drive. (It's a good idea to back up your Zotero folder to cloud-based storage or to an external drive this way regularly. This way, you won't lose your Zotero library if you can no longer access the computer you usually access your 'standalone' Zotero account on).
Be sure to close Zotero on both machines before copying the Zotero files from one computer to the other. On the new computer, copy the Zotero data folder to the same default location as the original computer or check the default location link above. 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.
Plan B! If you're regularly using more than one computer/laptop in your research, Zotero's sync feature can keep your library up to date on all of them. Zotero can store a copy of your library on the Zotero.org server and check it for updates whenever you open your library on a different computer/laptop. All your computers/laptops must be running the same version of Zotero Standalone.
First, set up your Zotero.org user account. 300MB come free (you might want to consider storage upgrade if Plan B is your backup preference). Then:
Repeat this configuration on each of your computers. Any updates you make on one of your computers will be reflected on the others. This even works to synchronize your library among Windows, Mac and Linux computers.