Cecily, We're doing this right now, with 2 boys ages 8 and 5 (drove Paris to Normandy then to Nice). I won't lie, it has been stressful at times. the roads aren't always marked or don't seem to match the Michelin map/directions I have. But the silver lining is that somehow, someway, we have always managed to find out destination. It has literally been by sniffing our way there sometimes! I'd recommend having a major city as a "drive towards" point so that if you get lost you can at least get back to a central location.
As for the kids...I hate admitting this, but the thing that quiets them down is the occasional promise of a McDonalds (and there are plenty of these), even if only for a milkshake. We've been here a month and they've been good about trying new things, so this is a reasonable compromise, in our minds.
In the end, I'm glad we have the car. It gave us a freedom and sense of adventure we wouldn't have had otherwise. And we've gone places we couldn't have gotten to on a train (or bus) easily, esp with the kids. Plus, it is REALLY nice to be off of public transport for a while (which presents other issues for kid travel).