The main page for downloading Zotero is https://www.zotero.org/download/. From this page, you can choose which version of Zotero you wish to download and install.
For a quick overview on using the Word Plug-in while writing your paper, look at the Zotero Word Processor Plug-in page.
You can insert Zotero citations into both Word and Google Docs.
If the Word Plug-in does not appear after installing Zotero, Microsoft Word may have been open during installation of the Zotero Word plugin. Save your documents and quit Word. Now, reopen Word. Your Zotero toolbar should appear. If not, try reinstalling the Zotero Word plugin with Microsoft Word closed–then reopen.
Both Zotero and the Zotero Connector should be updated automatically by default. However, you can always manually install a new version of Zotero over your existing version without losing any data.
The first thing you must do to use Zotero is register for an account. The free account option comes with 400mb of storage, though options for upgrading your storage amount are available.
Once you have an account, there are several ways to use Zotero:
Zotero 5: A self-contained desktop application where you can view and manage your bibliographic citation library. Zotero 5 regularly syncs file storage to Zotero's cloud servers automatically (when connected to the internet). You can manually add citations and attachments to your Zotero library, or you can import web-based sources by using the web browser-based Zotero Connector.
Zotero Connector: The Zotero Connector installs into your internet browser of choice and links the Zotero 5 desktop application to your Zotero library. Once installed, the Zotero Connector toolbar button will save your browser's current page (and certain file attachment types) to the Zotero collection you've selected in the Zotero 5 desktop application, as a bibliographic entry. In order to save an entry from the Connector, you must also have Zotero 5 open on your desktop.
Zotero.org web view: Zotero also allows you to browse your library through their website. Here you can find all the sources and attachments you've saved, and can organize your collection directly through your browser. Though you can manually add and organize entries through this method, you will not be able to automatically grab web resources this way.