To use Zotero most effectively (including the integrations with Google Docs and Microsoft Word), you will need to download and install both the desktop program and the browser extension. Once you have these you will still be able to use the browser version on any computer, but make sure you sync your account so that your citations are up to date no matter how you access them.
Below you will find a beginner's video tutorial to Zotero as well as links to additional video tutorials and Zotero's own documentation.