Family and friends, since I doubt you are all into skiing, maybe? Well I imagine you want to see the sights. So look may-september are the heaviest months, and lots of tourist. However April and October are hidden secrets. The flight costs less, the hotels cost less. However it is not as warm as in the summer. You can expect temps in the rome area to be between 50-75 F during the day and 40-55 F at night. If that is fine for you then those are the best months. Everything will be open, however less tourists so less lines, and less time wasted waiting to see something.
I would not miss Rome if i was coming to Italy, there are many good sites to see, Vatican, Spanish steps, museams, ect.
Have someone go to AAA before you leave and pay 20 dollars for an international drivers license. Then you can rent a car there. With a car, the gas prices are a little higher, but you can rent 9 seat vans, put all the family there and head to other cities, the cities are relatively close together 2-3 hours driving, and have tons of great sites. The van should cost no more then about 100 euro per day.
Have a good trip.