Don, Trenitalia will not accept U.S. credit cards. They don't tell you that - they just let you get the messages. All of the trains to Assisi from Rome originate in Rome Termini. They just happen to stop in Tiburtina on the way out of Rome. It looks like there are 2 Eurostar runs during the day from Termini to Assisi with no train changes. There are a few Regionale train runs that do this as well. All other runs will require a train change at Foligno. The Eurostar is a much nicer train than the Regionale but interestingly, the Regionale fare is 50% less and the travel time is 20 minutes longer. Trentialia won't sell you tickets for the R train anyway. But, don't worry. No seat reservations are permitted on the R train. From Rome Termini, it's an empty train. Just get there a bit early, validate your ticket at the little yellow box, hop on and find you own seat. Even if the train fills up (which it won't), you can stand. You'll be just fine.
Once at Assisi, you won't see the hilltown from the train station. Go next door to the little shop and buy a round trip ticket on the Line C bus. This bus goes to the top of Assisi. You will pass the Basilica on the way up (or, you can get off the bus if you want the Basilica first). The reason I suggest the top is that Assisi has steep streets. If you start at the top, you can casually and easily walk down. Going up all day long is not as nice. Once at the bottom, you can tour the great Basilica.
Note that the nice restaurants in Assisi close from 4-7pm.