If you already have some PDFs saved from past research, just drag and drop them from a desktop folder into your Zotero library. Zotero will automatically search online and match the PDFs with citations if possible.
Zotero automatically saves PDFs along with citations when possible (you can change this in the preferences menu if you wish.) Attachments take up space in your Zotero account -- you get 300 MB free and can buy more space if needed.
You can attach files manually by dragging a PDF (or other file) into your Zotero pane. Dropping a file onto a collection, or in between library items, will copy it into your library as a standalone item. Dropping it onto an existing item will attach it to that item. This is the easiest way to attach a copy of an article to its entry in your library.
Each item also has an Attachments tab in the right column. You can attach files by clicking the Attachments tab and then the Add button.