My favorites are The Woodyard and the original Joe's Kansas City. I wouldn't bother going to the other Joe's around town, they aren't as good as the original imho. I've heard great things about Q39, but haven't made it there yet. Jack Stack is good, but more of a nice restaurant feel than a bbq joint. I'm not a fan of A Bryant's or Gate's
Skip the Power and Light district. Young crowd and nothing special to eat or drink there. I've never felt unsafe downtown.
The Plaza is upscale, but has great restaurants and has beautiful buildings. I recommend going there one night.
Do the Boulevard Brewery tour. It's free, but you need to reserve your tickets online asap. It will fill up fast, especially in the summer.
As far as a bbq comp, I'm not really a fan of going unless you know someone competing. The only time I have gone without knowing someone is when I was researching smokers and it was great to talk to people about their smokers. People are usually friendly, especially if you go on a Friday night.
I personally like Stroud's, but don't eat a lot of fried chicken. It's worth going though. You get a ton of food.
If you want something besides bbq for a meal, there is a great little taco place called Taco Republic across the street from the original Joe's Kansas City. It has a great outdoor patio to have a few beers and the food is really good.
We love living in Kansas City. Nice size with plenty to do, but very easy to get around. If you are a baseball fan I would recommend going to a Royals game if they are in town. The stadium is one of the most beautiful in baseball. June in Kansas City is awesome. Not too hot and humid yet. Enjoy your stay!