If you are planning to use Zotero with a word processor such as Microsoft Word, the necessary plugins should be automatically installed when you install the Zotero desktop application. If this does not happen, or if you need to reinstall the plugins for some reason, you can follow the instructions on their word processor plugin installation page.
Since Zotero is open-source, there are additional community-developed plugins listed on their plugin page. Most of these plugins aren't of much use to the general user but browse the options to see if something may be useful or fun for you.
For mobile support, check their mobile page which has options available for both iOS and Android phones.
After you install the Zotero desktop app and the Zotero Connector, you will need to set up your preferences to allow for syncing and cloud storage of your citations. The first step is to register for a free Zotero account. You can do this on the Zotero website.
Once you have created an account, you can return to the desktop application and sign in. To do this, select Preferences under the toolbar tab Edit (on a Mac, select Preferences under the Zotero tab on the menu bar).
Next click on the Sync tab of the Preferences window. Enter in your username and password to your account and select both the Sync Automatically option and Sync Full-Text Content option.
If you would like to be able to access the attached files remotely, then select the Sync attachment files in My Library using Zotero. As of June 2022, Zotero gives all users 300MB of storage for free. Additional storage can be purchased on their website; however, other citation management systems such as EndNote offer more functionality if you are going to pay for the service. Most PDFs range in size from 100KB to 1MB, so Zotero's free storage can hold between 300 and 1,200 PDFs, depending on the size of the PDFs.
Additionally, this will allow you to access your library when you're away from your personal computer through the My Library section of Zotero.org. You can manually add citations using their website and it will sync with your library on your personal computer. The web interface for Zotero is not as powerful as some other citation managers, but it will give you basic functionality to add a citation if you have the citation information handy. So, if you're in a lecture and your professor mentions a paper by John Smith about something to do with Asymmetry published in Nature in 2003, you can add a citation on the Zotero website and tag the article with "follow up" or "to-do" to remind you to dig up the actual paper when you have some free time.
To access the stand-alone client (the desktop app) on your computer, open it from the programs menu. If you have not yet added the Zotero connector for your browser, you can do so from the Download Webpage.
The above image displays the layout of the Zotero pane. From left to the right the buttons are for: