As far as the highway patrol goes, you'll be traveling through Troop B from the Iowa line until you get to the Ralls/Pike county line. For the remainder of the trip, you'll be in Troop C.
Troop B is located in Macon, Missouri. According to the license, in addition to their main transmitter in Macon, they also have a 350 watt remote transmitter in Quincy, Illinois. They talk to the cars using 42.82 (PL 186.2). The cars talk back using 42.22 (PL 186.2).
The repeaters for Troop B cars really aren't worth mentioning because you won't be in the path of them.
You can hear car-to-car on 42.82 (PL 186.2), which is the same as the dispatch frequency.
As far as Troop C goes, I'll let some of the others answer your question for that area. I am not familiar with how it is divided up radio-wise.
For all troops in Missouri:
Car-to-car can be heard on 42.00 (PL 186.2), TAC 3 154.935 (PL 203.5), TAC 4 154.665 (PL 203.5)
For more information:
Department of Public Safety (DPS) Scanner Frequencies and Radio Frequency Reference