There are lots of promotions that run at Kauffman Stadium so be sure you check that out before you go. The Friday night fireworks are always great. They used to do $5 Monday tickets (who knows what the new season will bring). Sundays are family day and they often have a giveaway for the kids then base running after the game. My daughter's favorite promotion last season was when the KC Zoo and Build-A-Bear teamed up to give all the kids a free polar bear.
Parking is around $10. I always do promotions for tickets but have yet to find a way to get around the parking so be ready for that.
We really enjoyed taking our one-year old to a Kansas City Royals game. He enjoyed it too, but it was just slightly loud for that young of a child. It's a well-laid-out stadium with areas that toddlers can go to to run around a bit. Never overly crowded, we really enjoyed our experience there. Parking is not too bad either
Royals stadium is far enough outside of Kansas City that the traffic getting to and from games is not too bad. Tickets are easy to get and reasonably priced. Go on a Friday night in the summer for fireworks! Great place to take kids to watch a ball game.
Despite your stance on sports, this is a great bet for families. If you like sports, you get to watch a baseball game. If you like food, there is a ton of choices of sinful foods, and the cherry on top ... A huge kids area with a playground, baseball diamond, base running, and even a nice carousel. We aren't big sports fans but there were so many fun things to do that we only watched about 30 minutes of the game anyway. Tickets can get pricey but they run promotions all the time to make it more affordable and even gave the kids those little bandz at the gate.