I'm from Missouri and did two trips on the Katy Trail last summer. I rode from St. Charles to Hermann in one day, and camped in a park in town with campsites and showers. The next day I woke up and rode about 85 miles into Columbia Missouri.

Columbia is not on the trail, but the trail connects to it via a 10~ mile path heading north at McBaine that will take you into downtown Columbia. It's a great town, but you probably can't do it if you're planning on doing the whole trail in 3 days. It'll add about 20 miles on to go to town and back, but well worth it if you find the time.

I haven't taken the trail west of McBaine, so I will have to do that this summer.