I don't like Chicago Deep Dish, and prefer the tavern versions. But if you have to get deep dish, Lous is the best imo.
I would highly suggest going for a burger in Chicago at Au Cheval (or Small Cheval since you will be more north, and I think Wrigleyville has one). One of the best burgers I have ever had.
Saint Louis -- the best BBQ is not downtown. But if you cannot venture, the best imo is Bogarts Smokehouse. The rest are too modern/upscale, and have a real restaurant feel (like Sugarfire). They are delicious, but no old school pitmaster, etc. The 2 outside the city (about 30 min each). Dukes BBQ Shack and Ethyl's Smokehouse and Saloon.
Kansas City is EASY. Joe's Kansas City Bar-B-Que, and it HAS TO BE THE OG. The one in the Gas Station. Outside of a few spots I came to love in Raleigh/Durham when I lived there, this is consistently the best BBQ I have had in America. I like all sides, but never had their creamy slaw (spicy is good), favorite is dirty rice. Rocket Pig is my favorite sandwich, and probably the best item on menu -- though I am there so infrequently now, that I get meat by the weight and just get as much as I can eat (it is priced by the pound, you can usually get it at as little as 1/8 of a pound, though). 1000% get a piece of Texas Toast if your meal doesn't have it included.