Can I have two apple id’s and switch back and forth?

You can have and use a separate AppleID just for Game Center and/or the App Store and purchases if you want. Use the other AppleID for everything else (iCloud, FaceTime, iMessages). There would be no need to sign either in and out anywhere. Seems somewhat needless to me though. What do you expect to gain by using a separate AppleID just for gaming? And note that you can only change the AppleID used in the App Store once every 90 days.

Also, since your Game Center AppleID would not be the same as your iCloud AppleID, game data wouldn’t get backed up in iCloud.

In theory, each separate Apple service (iCloud, FaceTime, iMessage, the App, Music and book stores) could use a separate AppleID. But that seems needlessly complicated.

In theory yes, but is is not designed to be done that way and will not be practical. You will find it a nuisance with potentially combining your iCloud contents etc.