The first time the road will take you approximately 3,5 hours (230 km). We recommend renting a car with village roads and sharp turns in mind (not a small one; you do not need a jeep, a good sedan will be enough). Renting a car for the whole week will cost you approximately 12-15 thousand rubles.
First you will need to reach a train station in Rome (Termini or Tiburtina), and from there take a train to Assisi. You can find train schedules at www.trenitalia.it.