Accurate citations depend upon accurate data. Zotero, like any other citation manager, will produce citations based on the data it has, and if there are errors or omissions in that data, the citations will be incomplete or inaccurate. Spending a few minutes checking over and editing references as you import them will make your experience much better in the long-term.
Common issues with citations include stray characters and capitalization issues. Most of those you have to manually edit, but there is one that Zotero will help you fix. You probably know which type of capitalization (Title Case vs. Sentence case) your most frequently used style prefers. To make that change quickly, select the item you want to modify in the center pane. In the right-hand pane, right-click the Title of the source and select Title Case or Sentence case to have the title reformatted. You can do the same with the Publication name if necessary.
In most cases, you'll use the Zotero icon in your browser bar to add references to your library. When you're looking at something in your browser that you want to add, just click on the icon in your browser (and make sure the Zotero program is open).
The icon that displays will depend on the type of item on the page. For example, it may show up as:
a bright blue page for a website or magazine article...
or a book or a journal...
or a page with lines for a journal article.
When there are multiple references, like on a page of search results, you'll most likely see a folder icon.
Click the icon to bring up a box that will let you select a single item or to select items from a list.
If you have folders in your library, a little pop up box near the icon will ask you where you want to save your item/s.
When saving web pages to your library, Zotero will save the citation as well as a snapshot of the page (unless you've turned this off in your preferences). If there's a PDF file on the page you're viewing, or you're currently viewing one in your browser, Zotero will import the PDF as well as the citation.
You can save items to your library directly from your mobile device.
iOS
On iOS (iPhone or iPad) install the Zotero App 
and use the Share button 
to access the Share sheet.
If you are continually prompted to login to your account, but items still don't save, go to your Settings, find Safari, and turn off "Prevent Cross-Site Tracking."
Android
Zotero is in the process of developing an app for Android mobile devices. Until then, you can use one of these options:
1. Save webpages or identifiers to your library using the "Save Page" provided by Zotero.
2. Access your Zotero web library in any browser on your Android device. Navigate to the web library. Turn your device to the landscape position. Select the three dots in the upper right corner.
Select "Add by identifier."
Paste or type in the URL, ISBN, DOI, PMID or arXiv ID and the item will be added to your library.
If you use the Browzine mobile app:
In iOS use the Share button and select Zotero to add the citation, and the PDF if available, to your library.
In Android, select the three dots and then select "Send to Zotero" to add the citation, and the PDF if available, to your library.
If you use the web version of BrowZine, use the graduation cap icon to add the citation to your library; you'll have to manually download and drag and drop the PDF to obtain it.
Within the Zotero client, there are a few ways to add items:
