Im personally going with postgresql, no specific reason. I dont think really matters too much what you database you go for, just stay away from document/noSql databases.
Interms of hosting, when im going into productions, im going to be using digital ocean managed db service, its really really simple to use and connect. (not free, about 5-15$ a month, i think GDPR complaint as well)
For development, ive specifically gone with oracle server, its 4 cpu 24gb ram server that is completely for free, that server will be able to handle anything you can throw at it. And then just running docker on it, running a docker database on it.
The above oracle server, wont be easy, but i did it for learning security, ports, setting up proxy managers and so on. Thats if you want to learn more infrastructure side but its not necessary.
If your after a free database for development, you can try free here on:
or (mysql (and some limtiations))
But there are so many databases as a service now popping up.
Also for local development you can setup a local db on your computer, but i dont like to do that as i like to always have idea of how long queries will take and just getting used to using a database remotely.