You have to try Gates or Arthur Bryants to get some good KC BBQ. But since you will be out in OP you can try Jack Stack, they have great BBQ and its a little more fine dining than some of the other BBQ joints.
You might also want to check out the new Legends shopping area out by the Nascar track. Besides the BBQ joints there is not a lot of thing specific to KC that you wont find anywhere else in the US.
I dont know if I saw what time of year you will be here but if it is after thanksgiving you should check out the plaza, they have it all done up for Christmas and there are tons of people out walking around, its a park and get out kind of place. I think it was the first "shopping center" in the US. If im wrong someone will correct me.