I would second Ed's pick for Arthur Bryant's BBQ (inside Ameristar, and my favorite poker room in KC)- just to put emphasis on how superior Bryant's is to the other KC BBQ joints.
runner-up BBQ would be Fiorella's Jack Stack. As for other things to do/see- I recommend the Plaza in downtown KC. Nice outdoor shops and restaurants to experience. Can't beat a cheap Royals' game either! I lived in Overland Park and Olathe for 2 years- definitely miss Kansas City!