You always play first in the first turn. After that it depends on who passed first when both sides pass. When a turn ends, it's because both you and Gary have passed. If you pass then Gary passes, you get to go first in the next round. If he passes then you, then he gets the initiative.
I utterly failed to understand the original question, considering even I knew that! skip_intro said it perfectly.
Yup, there's an awful lot of strategy in passing. There's a huge advantage to passing first since you get to play first with fresh new cards after the draw, but sometimes you're better off playing out your hands instead. Also, you can't always be sure whether your opponent is going to pass or keep playing. A nice strategy, if your opponent has already passed, is to goad them into playing something. Moving a warrior up to a squishy wizard, for instance. Then they play, you pass, and you get initiative at the start of the next round.
The first player in the first round is selected randomly in MP, but from then on it follows the same passing rules. There can be an awful lot of bluffing
That's actually really clever. I always thought Gary was just a derp when he passed early. I'd attack after he passed and he'd attack back and I'd be like "well why didn't you do that last turn?" BUT, could there be some sort of feedback then? Put "Gary has passed and will go first next turn" on the screen. Or at least "Gary is going first because he passed first" in the log.
Yes, some kind of indicator for this would be a good idea. I already knew how the turn order worked from reading the forums but otherwise it wouldn't have been that obvious.