Once Zotero and any appropriate connectors are installed, you can begin collecting information about sources. Zotero "senses" information on a web page, and if it understands the data, easily lets you capture citation information into your library simply by clicking the icon in the URL/location bar.
Do you want to attach PDFs to citation information and store those PDFs in the cloud? There are two ways to do that.
If Zotero can sense the PDF on a citation page, you can set your preferences so Zotero automatically downloads the PDF. Click the gear, then select Preferences.
Choose General and make sure the box next to "Automatically attach associated PDFs and other files when saving" is checked off.
Then when you save a citation the PDF will automatically be saved with it. (Note: if you're using the article database JSTOR, you'll have to download your first citation manually in order click "OK" to JSTOR's terms and conditions. All subsequent JSTOR citations that you save during that session should download automatically.)
File didn't automatically download when you saved the citation information? No worries. You can simply save the PDF to your computer, then right-click on the citation in your library and attach the file to it.
Zotero will move a copy of the file to your Zotero storage folder on your computer AND will sync it online (assuming you set up syncing of files earlier.)