In addition to F-Droid, you can get Aurora Store (which is on F-Droid) which lets you use an anonymous login to get at the Play Store. I use it for a couple free software apps that aren't on F-Droid for some reason.
The only slight dowbside is that I haven't figured out how to auto update appd, so your apps will get out of date without you being notified and you have to manually do it. This problem might literally be solved by a simple setting thay I haven't bothered to look for, IDK.
On the plus side it includes all the official play store apps, along side some that aren't allowed by play store.
For examples, Newpipe, the superior replacement YouTube app that isn't allowed on play store due to it subverting advertisements and allowing a few features that are useful for downloading certain things.