Zotero consists of the Zotero program itself, a Connector that allows Zotero to connect with your browser of choice, plus optional third-party plugins. Start by downloading and installing Zotero and the connector for the browser you use most often: Chrome, Firefox, or Safari.
If you wish to sync your reference data to your library on Zotero.org, or if you will be using your library on more than one computer, you will also need to register for a Zotero account to enable data syncing. This is free and includes unlimited storage for references, plus the option to enable file storage for PDFs and other attachments. (300 MB of file storage is free, with more available for purchase.)
Once you have installed Zotero and your Connector of choice and have set up your Zotero account, navigate to the Preferences tab in Zotero and enter your login information to enable data syncing. Now you're ready to start collecting references!