This guide will take you through the basics but there are many resources available at Zotero.org to help you learn Zotero. My advice is to install it and jump right in. Then, when you can't figure something out, check out Zotero's Support Page and Video Tutorials.
For detailed instructions, visit the Getting Started Checklist.
There are many plugins available from the Zotero community that can be found at Plugins for Zotero. When you download the Zotero desktop application you will have the option to enable the most popular plugins.
This is important for saving and syncing your citations. You need to do this step for each computer with the Zotero desktop application and for each Zotero connector if you use more than one web browser. Once signed in, you should not have to do this step again.
In the Zotero desktop application:
In the Zotero connector:
There are videos for the Zotero connector setup available on the Zotero Connector Setup page.
Zotero is not cloud-based. What does that mean?
It means that you will need to sign in to your account when using the Zotero Connector and Zotero.org in order to save your citations across multiple computers. As long as you are signed in, syncing should happen automatically when your start the Zotero desktop application.