I use an 8bitdo SN30 Pro + 8Bitdo Smartphone Clip together with emulators on my phone to enjoy Gameboy, SNES and PSX, I’ve also tried emulating some newer consoles but my phone is a bit weak for that.
There’s a ton of great controllers to turn your phone into a proper gaming device, I chose the 8Bitdo because I can also use it with my computer over Bluetooth.
But if that isn’t a consideration then I would probably go for one of those that has passthrough USB-C Charging so you can charge both controller and phone while playing.

Before I got the controller I mostly played Gameboy Pokemon games on my phone.