You can organize your library by creating folders, which are called collections in Zotero. You might find it helpful to organize items by theme/topic or by a specific research project you are working on. You can create a new collection by clicking on the "New Collection" icon or right-clicking on "My Library" and clicking on "New Collection."
You can also create subfolders by right-clicking on a collection and clicking on "New Subcollection." Right-clicking on a collection will also give you the option to rename or delete a collection.
You can add notes and tags to items in your Zotero library. There are tabs for notes and tags in the right-hand pane next to the "Info" tab.
Tags provide another way for you to organize your library. You can search by tags using the "All fields & tags" search box. Some databases will automatically provide tags (or keywords) when you add an item to your library. You can disable automatic tagging by going into your Zotero preferences.
Zotero has an integrated PDF reader that allows you to read and annotate PDFs within Zotero. To read a PDF, double-click on the PDF you want to read and it will open in a new tab. From here, you can highlight or add sticky notes to the PDF as you read. You can change the color of your highlights or sticky notes if you want as well.
If you want to make your annotations searchable, you can add them as a note (either individually or all together). First, click on the notebook icon in the right hand panel.
Next, click on the plus sign next to "Item Notes" to either create a note that you can then add individual annotations to (by right-clicking on an annotation and selecting "Add to note") or by creating a note containing all of your annotations from the PDF.
Once you've created your note, you'll be able to click on the citation at the beginning of the note and select "Go to page" to find your annotation within its original context.