Obviously, you'll get this on the way to these two: -
but how you play through these early Game Shows depends on whether or not you're going for all the other mini-game related achievements at the same time. I'd recommend going for all those as early as you can so they're out of the way.
But, if you just want to get this achievements as fast as possible (which would be a bit of a waste) then you can skip chunks of the game by mashing A. You can also skip the AI's turn in the Yahtzee Bowling and accept whatever you get on the first bowl. For Bop-it you can just do nothing. Connect 4 Basketball, Boggle Flash and Sorry Sliders is usually over quicker if you try and win.
Unfortunately, there's no way to speed up the end 'game' with the cash cards so you'll just have to sit through it. Each full Game Show takes about 12 - 15 minutes to get through.