We will be in Rome and riding the train to Florence. Does the direct/fast train stop in orivieto or do we need to buy separate tickets?
I don't believe the high speed Freccia trains stop in Orvieto, however some of the Regionale Veloce trains do. Are you planning to stay in Orvieto for a few nights, or just visit for a few hours? You can check all the rail solutions at http://www.trenitalia.com/ .
If you want to visit Orvieto, you would need separate tickets. If you are traveling on one of the high speed trains in Italy, you have a reserved seat so you cannot hop off, hop on. For an intermediate stop, you'll always have to buy the appropriate tickets.
The direct fast train does not stop in Orvieto. It does not go even through the station in Orvieto. It has its own separate tracks. You will probably need tickets Rome - Orvieto and Orvieto - Florence.