Carroll, my preference would be to use the train, as it's much less "complicated" than having a car. At the moment, fuel costs are a concern, not to mention parking, tolls, the dreaded "Zona Traffico Limitato", etc. With a family of five, a car may be more economical, but especially as you're traveling in "peak" season, the other factors should certainly be considered.
One possibility might be to use train for most of the trips, but rent a car for the day trip to Modena. Note also that you'll require an "International Driver's Permit" to drive in Italy, which MUST be used in conjunction with your home Driver's License.
This topic seems to come up on a regular basis on the Helpline, so you might want to do a search to view some of the other opinions as well.
Cheers!