I understand that not all games will be compatible with Wine, but how do I start a game in Steam while running Steam on wine? Do i just press play?
If a Steam game is able to run with Wine, and you're running the Windows version of Steam with Wine, then yes, double-clicking on the game in your game library or pressing Play will (attempt to) run the game with Wine.
This is really an instance of a more general question: What happens when a Windows program running with Wine tries to launch another Windows program; does Wine try to run that, too?
The answer is yes. Wine supports the parts of the Windows API that Windows programs use to launch new processes.
With that said, it is possible that you could have some Windows Steam game that only sort of works with Wine on Ubuntu and requires special steps to launch. In that case, you might have to do something different from just launching the game in your Steam games library. But what you'd have to do would be particular to the specific problems running the game. In general, to run a Windows Steam game from the Windows version of Steam in Wine, clicking Play is the way to do it.