Zotero is an open source program, so it is completely free to use.
The Zotero for Word plugin should be downloaded automatically when you downloaded Zotero.
If you don't see the ZOTERO tab when you open Microsoft Word, try this:
Edit -- Preferences -- Cite -- Word Processors
If you want to attach full text articles in Zotero, right click on the source in your list and "Add Attachment".
Also, if the URL is missing from the citation information in the right sidebar, just click in the blank space next to URL and copy and paste it there.
To get the citation for a source you've put in Zotero,
You cannot download the desktop version of Zotero on a Chromebook. What you can do is use the online library:
(1) Register for a free account, then log in
(2) Download the Zotero Connector.
This will add a Zotero icon in your browser toolbar.
(the icon changes depending on what resource is on that page)
(3) Authenticate the Zotero connector
(Right click on the Zotero icon in browser toolbar --> Preferences --> General --> save to Zotero.org --> Authorize --> Accept Defaults)
(4) You may get an error pop up. If so, just click the button for "...online library"
That's it! Now you will be able to send items to your online library.