Looking Up Train Schedules and Routes Online gives you the DB train schedule link and tips for using it. This is just to view schedules. To buy tickets, you can go direct to voyages-sncf.com (credit card approval may not be easy) or use an agent like RailEurope.com or trainline.eu.
You'll have to indicate whether you prefer (are leaving your car near) the Belfort Ville (town center) station or the Belfort-Montbéliard TGV station. Trains from Belfort Ville depart every two hours with a connection at Mulhouse. Direct TGV trains departing the TGV station are less regularly timed, but still several per day.