Here's how you travel easily with kids and luggage on the trains:
When leaving your hotel, have them call a taxi. It will come to the door in a few minutes, and will have a few euros on the meter already (they start the meter when they get the call). Tell the driver you want to go to the train station (in cities like Rome, make sure you know which one).
When you arrive at your destination, go out the front door of the station to the taxi rank. Ignore ALL offers of taxis from anyone else. The lines move quickly. Have your hotel's full name, address and phone number written out, and show the driver (this way there's no language barrier, and he can call the hotel if there's an issue).
You can't drive in city centers with your own rental car due to the ZTL's, but taxis can. So, taking the train and using taxis is actually the best way to get door-to-door service (which I agree is indeed a plus with children and luggage, and well worth the extra cost compared to buses or walking from the station). If you drive yourself, you have to park on the outskirts and take a bus or taxi anyway, so the train is no less convenient.