Tha main train station is some way from the town centre, regular bus service or a short taxi ride.
There are rail links to Munich, Augsburg, Ulm, Lindau and Oberstdorf that we saw, probably more as well.
This makes Kempten a good place to explore the Allgäu region from.
Over the day each hour there is a train connection from Munich to Kempten. You can find the times and prices on the homepage of the german train company (Deutsche Bahn). See the link. Enter 'Kempten Hbf' as destination. It is the main station. Unfortunately the trainstation here is not downtown, so if you have much to carry it is better to take a taxi or the bus (ZUM) to get downtown. Sorry, the local traffic page is just in german.