Zotero will let you send PDFs to/from a tablet with the ZotFile (http://zotfile.com) plugin. This works well if your workflow is (1) find PDF on your computer; (2) read on tablet. But if you want to do anything else -- even choosing a PDF to read from your tablet without touching your computer -- then Zotero won't work. With that said, I know Zotero is working on mobile apps...not sure how far they have come though.
If you're Mac/iOS only, also check out https://www.sonnysoftware.com/bookends/bookends.html.
Also, Bookends annoyingly checks to see if you are running the same license on multiple devices on your network, and if you have more than once instance running it forces you to close it down. This may seem superficially reasonable, but for someone with both a desktop and a laptop it's quite annoying. I emailed the developer about it and they didn't seem to get why this was a problem.
I like Zotero, but the lack of an ipad client is annoying. The file plugin requires more forethought than I really like. Bookends having a mobile client was one of the attractive features.
What is the work that it takes to get set up?