Zotero's greatest feature is its ability to automatically recognize what item(s) you're viewing on a webpage and bring them into your reference library. The Zotero connector icon you click to save citation information will change depending on what kind of resource you are viewing.
Adding Citations Manually
Start by selecting the green "plus" icon and pick which kind of resource you want to add.
You will then have the ability to manually enter the reference information for the source you plan to cite.
Organizing Your Library
When using the Zotero desktop application, you'll see a folder with a green "plus" near the top left of your screen. Click this to create a new "collection."
You can create collection to organize your reference. Think of collections like file folders on your computer, but a reference can belong in more than one collection at a time.
Saving a Citation with One Click
The Zotero icon lives to the right of your browser's address bar and changes depending on what kind of resource you are viewing. If you want to save the citation information, just click that icon.
Zotero will automatically pull all of the citation information into your desktop application.
You will then be able to find the resource downloaded to your Zotero library.
Saving Multiple Citations
If you're on a page of search results with many items, you'll see a folder icon instead. Click it to get a list of all the items on the page, and check off the ones you want to save.
Attaching a File
You can easily attach files (like PDFs) to references in your Zotero library. Just drag the the file into your Zotero pane. Dropping the file into a folder will create it as a standalone item. Dropping the file into a reference will add it as an attachment.
(Remember that you get 300 MB of free storage if you're syncing your library onto Zotero's online servers. If you're only storing your library on your own computer, you don't need to worry about storage limits.)
You can also directly attach to items by right clicking on a reference, which will bring up several attachment option. If you want to have the actual file attached to a reference, use the "Attach Stored Copy of File..." option.