I'm from Kansas City. I highly reccomend Jack Stack over AB. Go to the JS that is downtown, off of Broadway. It has better food, and is not in a really bad slum area. Enjoy the food and the game.
Gates is easy enough to go to, it is right on Main St. at 31st, and they have good food. I never liked going there though because the service is attrocious. They literally yell at you everytime they speak to you, and it makes you feel unwelcome. The place is usually pretty dirty too. I guess it would be alright for a one-time experience though.