Can we rent a car at the Rome airport and then head out to Assisi? Or do we need to take a train to a rental location?
There are rental locations at Rome airport as at any other airport. Make sure every driver has an International Drivers Permit.
Yes, of course you can rent at FCO airport.
Consider renting thru a consolidator such as www.autoeurope.com or their affiliate www.kemwel.com.
Your pickup location is Rome airport. If you intend to visit Rome in the last few days, return it at a location near the Tiburtina station.
If you have never rented cars in Italy, the above linked websites contain all kind of info on car insurance in Italy (most credit cards don't cover Italy), international Driver's permit, etc.
Also Google and read info about European (or Italian) road signs, and (very important) Limited Traffic Zones (ZTL) in Italy's historical centers of most cities and towns. Access to motor vehicles is restricted to residents in those ZTL areas and enforcement is by "candid" cameras (not live cops).
There is no cruising in the left lanes of Italian freeways. The left lane is passing lane only and you must clear it for faster vehicles as soon as you are done overtaking a vehicle.
Thanks for the advice. I will definitely look at the websites for more info.