A friend from work is a KC native, so I've gotten to go to Kauffman recently.
As one of the other posters mentioned, there is a kid's area in the outfield that looks pretty cool. If you get a chance to tour Kauffman, by all means do it. Gorgeous, clean ballpark and the fountains in the outfield are amazing.
The Plaza area in KC is very nice for shopping and dining. It reminded me of Oak Brook or Orland Park.
As for barbecue, my friend got me hooked on Jack Stack. They have an entree with your choice of two meats covered with BBQ sauce. My mouth is watering just thinking about it. Excellent.