In order for you to send PDFs to your mobile device, you'll need to connect the Zotero software on your computer with the folder on your computer that syncs with a folder on a cloud storage platform. Complete instructions are available on the ZotFile site.
In Zotero's Tools menu, select ZotFile Preferences. Then, in the General Settings tab, choose the folder that syncs with your cloud storage.
Then, in Tools > ZotFile Preferences, select the Tablet Setting tab and choose the folder on your computer that syncs with your cloud storage. If you check the box to rename the annotated PDF file, please note that this will add an additional PDF to your Zotero library instead of replacing the PDF. Consider whether or not you'll need an annotation-free PDF in your Zotero library before selecting this option. More info from UC Berkeley LIbraries' guide on Zotero and ZotFile.
When you right click on an item in Zotero, you'll see additional options under "Manage Attachments":
Sending to Tablet transfers the PDF to the folder that syncs with the cloud (and thus the folder on your device(s)).
Once you are done annotating, you can then select "Get from Tablet" to retrieve the annotations, which will come over in a child note in Zotero:
Remember to consider price, long-term access to files, data privacy, and support across devices before signing up for any of the following:
Box: (free storage limited) Use across a variety of devices.
Microsoft OneDrive: (free storage limited) Use on Windows/Mac/iOS and Android. More support available on the Microsoft website. Note that you can use OneDrive to annotate on a Kindle Fire.
Google Drive: (free storage limited) In order to sync your PDFs across devices, you will need to set up Google's Backup & Sync tool. If you want to read & annotate on a Kindle Fire, please note that you will need to sideload the Google Play store in order to use Google Drive on your Kindle Fire (more info from The Verge)
Dropbox: (free storage limited) Use on Windows/Mac/iOS and Android. More support available on the DropBox help site.
Amazon Drive: (up to 5 GB of storage with Amazon account)
pCloud: (may offer free trial with 10 GB of storage) Use on Windows/Mac/iOS/Linux and Android. More support available on the pCloud Drive Help site.