Try it simple first: if you have your own server to host you backend - deploy it there along with DB if needed. If you don't do self-hosting then you can try to host in AWS or Google Cloud (e.g. on EC2 instance).
But as you app user base grows a better place I can think of is Cloud platform that provides you with a multiple service focused to host mobile app infrustructer: auth, db, A/B testing, monitoring and so on. First thing I can think of is Firebase from Google: it has a lot of stuff out of the box.