For BBQ in KC you need to check out Oklahoma Joe's, Arthur Bryant's, & Gates.
Oklahoma Joe's is awesome. If my memory serves me correct, it was on one of the food network shows as one of the top 10 places to eat in the world. The only thing is, it's in a gas station!
Arthur Bryants is a well known BBQ joint that people travel from all over to get. Their sauce has a strong vinegar flavor to it that I don't care much for, but everyone else I know loves it. The meat is great though.
Gates is one of my favorites, and their extra hot sauce is my absolute favorite sauce. It's what I use on most of my Q, both the rub and sauce.
Whichever one (or all) you choose, go hungry. And they will all usually have a line out the door.
And if you need something to wash it down with, I like the Flying Saucer in downtown KC (Power & Light District). They have a good selection of brews, and it is fairly close to Arthur Bryant's if that helps your plans at all.
Have fun!