Welcome in advance to Kansas City! In case you aren't aware the Missouri side is the biggest part of Kansas City. If hitting up breweries is your goal then Boulevard, KC Bier Company, Martin City Brewing and Cinder Block Brewing are musts. If you go to Cinder Block you might as well go to Big Rip as its just five minutes away.
Since Boulevard restructured the way they do tours it's actually fairly easy to get in, especially on weekdays. They have three types of tours, the free tour and two levels of paid tours. With the free tour you get three complimentary sample beers and AHA members also get a free pint and 10% off merchandise. You still need to get there early in the morning (9 or 10 I believe) when ticket vouchers are available, but you can get the vouchers for a later tour and come back when it's time. If you've seen enough breweries you can just go to the Boulevard tasting room to drink and stare at the brew system through a window. KC Beir Co, in the Waldo neighborhood of KCMO, specializes in traditional German ales and lagers. Their Weizenbock is as good or better than Weihenstephaner Vitus IMHO. As far a locally owned brew pubs there's McCoy's Public House in Westport and 75th Street Brewery in Waldo. They both make great beer and pub food but I give McCoy's a slight edge for their Pot Roast and Mac N' Cheese.
As far as beer bars I'll skip the chains and only mention local places. In Westport there's The Foundry, Beer Kitchen, The Riot Room, Green Room (also a nano-brewery) and The Local Pig (formerly Bridger's Bottle Shop). On the Kansas side out in the burbs there's Barley's Brewhaus and The Blue Moose.
I highly recommend seeing the Royals as they are an exciting baseball team and play in what many consider one of the nicest stadiums in baseball. There's also a Drury Inn across the street which has three complimentary adult beverages from 4-7PM.
I'd stay away from The Legends in KCK unless pro soccer or NASCAR is on your to-do list. While it has great restaurants as the previous poster mentioned, it's just like any other suburban shopping district in the US. To get a true sense of Kansas City I'd stick to the following neighborhoods/areas (north to south): The River Market, Downtown/Power & Light District, The Crossroads Art District, The Freight District, Midtown, Westport, Brookside and Waldo. Also, KC North if you hit up Cinder Block and Big Rip and SOuth KC for Martin City Brewing Co. As far as where to stay I'd stay in downtown KCMO or on the Plaza. There's also hotels in Westport but that's a late-night bar area so that may not be good if you like go to bed early.
The best BBQ is hotly debated but according to Anthony Bourdain Joe's is one of the 13 places you must eat before you die. He said it's "the best barbeque in Kansas City which means it's the best barbeque in the universe." Jack Stack, Rosedale, Danny Edwards, RJ's and Q39 are also great places to eat. Gate's and Bryant's are okay but they're the stereotypical sauce-covered KC BBQ. Good non-barbeque upscale casual restaurants are Blue Stem, Cafe Europa, Plaza III (best steaks and good happy hour), Webster House (great happy hour) Le Fou Frog (French), Lidia's (Italian), and Micahel Smith (fine dining). More casual are Ponak's, Rudy's and Manny's (all Mexican), Jazz Louisiana Kitchen, Po's, Lulu's and Blue Koi (those three are noodles), and You Say Tomato (breakfast brunch).
You also need to visit the Jazz Museum and the Negro Leagues Baseball Museum. Both are next door to each other and a few blocks away from Arthur Bryant's Barbeque in the historic Jazz District. The World War I Museum across from Crown Center is also an awesome museum.
I hope that helps! Enjoy your visit to Kansas City!