Zotero works best for most people when you download the desktop app and use that as your frontline Zotero, then syncing to your online library.
However, there may be circumstances in which you have no choice but to work online only, without the desktop app. Being a Chromebook user is one of those circumstances for most people -- Chromebooks generally don't allow installation of desktop apps.
If you are working remotely with a significant number of students (during the COVID era, for example), getting everybody on board and troubleshooting individual machines with Zotero desktop app may be difficult -- you will probably be better of using Zotero in the cloud.
So this page will walk you through the process of using Zotero without the desktop app. If you are using the desktop app, you should not bother with this page.
[This page assumes that you know the basics of saving citations in Zotero]
Saving and Organizing Citations
All citations will always be saved to My Library by default. This is true for both the desktop app and Zotero in the cloud. However, there is an important difference. In the cloud, citations will always be saved only to My Library. In the desktop app, you can indicate that you also want the citation to be saved to another specific collection.
To move a citation from My Library to another collection, you can drag and drop, or use the icon for moving items to different collections:
Saving Full Text PDFs
In the Zotero desktop app, you can set your preferences to capture full text pdfs (where they exist) along with your citations. In the cloud, it appears as if this happens by default, but so far as I can tell, the full text PDF is not actually saved, only a placeholder for it is. You can easily upload a copy of the PDF that will become an attachment to your citations, though. Just be aware that you will rather quickly use up your free server space on zotero.org if you begin collecting pdfs.