Download and install: Go to http://www.zotero.org/ and Download.
Download Zotero, but also the Connectors (plug-ins) for Firefox, Chrome, Opera, and/or Safari.
Note: The version of Zotero that worked within Firefox is being discontinued.
You will probably need to allow a Pop Up. Then restart your browser after you have installed the Zotero Connector.
Set Preferences.
Click on Zotero icon, usually in lower right.
Look for the Gear icon on top of Zotero panel. Touch arrow and select Preferences.
General: Click all boxes under Miscellaneous.
Sync: At some point, set this to work with your Zotero.org account. I suggest that you not "Sync Automatically."
Advanced: "Search for Resolvers" and "Show Data Directory." The location of your Zotero files is important. You will need to back up those files at intervals, especially the "Storage" file.
Near the bottom, the resolver (Sept. 2014) should be: http://resolver.ebscohost.com/
Zotero Preferences information and video.
Word Processor PlugIn: Go to the Word Processor PlugIn and install that too. It's very useful for adding citations as you write..
Open the Zotero panel, using Zotero icon in upper right. (The Zotero panel will open up to about half of your screen. You can change its size. I usually close it quickly, since it can take up lots of screen space and it's easy to reopen.)
When the panel comes up, click on the file or "New Collection" icon at far left. Name it.
You will then see your new file under My Library. You will be able to drag citations from the middle column into the file.
Right click on the file and "Create Bibliography from Collection." Choose your citation style. I usually "Copy to Clipboard" and paste into a document.
Look for Zotero-created icons in the far right of the URL box. Click on the icon and Zotero will almost always be able to collect the citation.
Article
Book
List of results
Click on these icons and Zotero will collect the information. For articles, Zotero will also try to collect the full text PDF (if there is one) and a snapshot of the database record.
If Zotero does not recognize what you've found, so that you do not see such an icon, you can also "Create New Item from Current Page." (Sometimes when I cannot get the easy icon, I'll search for the same article in Google Scholar or the same book in Amazon or WorldCat. The icon will usually be there.)
And you can drag a just-saved or previously saved PDF into the middle part of the panel. If you then right click on the item, you can click on "Retrieve Metadata for PDF." The Zotero will try to extract its information.
Click on Gear icon, then Preferences and Advanced. The click on "Show Data Directory."
Then close Firefox (or standalone Zotero). Save the zotero.sqlite file to safe storage.
Syncing is not a reliable way to back up, because you can sync over valuable citations unintentionally.
Zotero will try to collect PDFs, but also, you can drag a just-saved or previously saved PDF into the middle part of the panel. If you already have a citation, you can drop it onto the item.
You can also link to a previously saved PDF with "Attach Stored Copy of File," as seen here:
If you do not already have the citation information in Zotero, then right click on the PDF line and click on "Retrieve Metadata for PDF." The Zotero will try to extract its information.
If Zotero is successful, you will see the information in the rightmost column of the Zotero panel.
If you have everything but the full text, use the green arrow "Locate" button to find the full text. This will work best if "OpenURL" in Preferences/Advanced has http://0-linksource.ebsco.com.wncln.wncln.org/linking.aspx . See above under Setting Preferences.
Use the "Add Note" icon to attach notes to your citation.
If you have not done so already, go to the Word Processor PlugIn and install that.
In Microsoft Word, you will find the Zotero functions in Add-Ins.
The leftmost icon 
will insert an in-text reference, and the third one 
will insert a list of endnotes.
On the other hand, you can usually drag and drop from Zotero into almost any word processing software.