Zotero has two major ways of allowing you to save full text along with your citations.
PDFs: When a PDF of an item is available (in a library periodical database like JSTOR, for example) Zotero will by default grab the full text PDF and attach it as a child to its parent citation.
If you don't want full text PDFs, you can turn this feature off under Edit --> Preferences --> General. Uncheck "Automatically Attach Associated PDFs and Other Files When Saving Items".
Snapshots: By default when saving an item from most websites (including, for example, blog sites, major news sites, magazine sites, etc) a snapshot of the page will be saved along with your citation. In most cases this will serve as a full text copy of the item.
If you don't want full text snapshots, you can turn this feature off under Edit --> Preferences --> General. Uncheck "Automatically Take Snapshots When Saving Items From Webpages"
Note that if you are synchronizing your Zotero library to the free space provided at Zotero.org (and you should be), synchronizing attached PDFs and snapshots will use up your free storage space fairly quickly. To solve this problem, you can
You can highlight and annotate PDFs in your Zotero library via Adobe Acrobat Reader. OSU has a good page on this here. These annotations will not sync well however.
You used to be able to annotate snapshots, but no more.
You can add PDFs to your Zotero libraries on their own by dragging and dropping. If you drop a PDF onto a citation, it will become a child of that citation. If you do not drop it onto a citation, it will stand alone as an orphan PDF with no bibliographic information associated with it. Zotero can often retrieve full bibliographic information for an orphan PDF. To do this, right click on an orphan PDF in your Zotero library and choose "Retrieve Metadata for PDF". If Zotero can retrieve the metadata, it will create the relevant item, fill out the relevant bib fields, and attach the previously orphaned PDF as a child item. (The first time you do this you may be required to download relevant tools which is free and easy. On campus machines, you may have to do this every time).
Where possible, Zotero will associate full text PDFs and snapshots with their citation. An item that has an associated child item will have an arrow just to the left. When the arrow is putting to the right, the child items are hidden. When it is pointing down, you will see the child items listed underneath the parent. You can change the arrow direction by clicking on it.
Parent items are usually citations. Child items are usually full text PDFs, full text snapshots, or notes that you have written. When you move a parent item, child items go with it.
Here is what it looks like in your Zotero main panel when your parent items are closed. The Journal Article and the Web Page both have right-facting arrows, indicating child items. The book does not (which is typical for books):
And here is what it looks like after you have clicked on the right-facing arrows. The arrows are now down-facing, and the child items are visible. In this case, the full text PDF of the journal article that came from a scholarly database, and a full-text snapshot of a webpage that came from a site that Zotero couldn't recognize :