I tried something similar for a while. I had an OpenVPN server set up at home and would connect with my phone, so all my phone's traffic was behind the same protections as my home network. The main issue I experienced was when going through areas without cell service (which happens frequently where I live). I would have to restart my VPN client in addition to any apps that required internet connectivity. Not the safest prospect when you're streaming music and doing 70mph down the highway.

At this point, I don't have anything to protect my phone's traffic when I'm away from home. However, essentially every app on my phone is a FOSS package, so I think the biggest risk to data privacy is probably Google.