You may save a few quid driving, and I think it would be quite nice if it was just adults travelling, but with two kids in tow I think it would be worth paying for the flights (just imagine the first “are we nearly there yet?” coming an hour or so into the journey!).
I think Spain is a good choice if you are going general admission. GA is pretty good anyway in Spain, and with the recession hitting their economy hard, it won’t be too busy. Italy is a fantastic circuit, but there is lots of walking from the nearest entrance to the track and there isn’t as much good GA viewing. If flights are too expensive, and you have to drive, I would put Spa on your list – a much shorter journey.
Back to Spain, it is pretty easy to do everything independently. Another option is to arrange your flights & transport to your hotel yourself, but book GP tickets, Hotel, & circuit transfers as a package. We have booked for 2013 through http://www.montmelotickets.com You can stay in Barcelona, or for a cheaper option, with shorter transfer time around 30 – 45 minutes hotel to circuit. (& maybe better for the kids as the hotels have pools & some have entertainment etc in the evenings) take a look at their Costa Brava deals: 3 Nights B&B, GA ticket & Circuit Transfers from €256 / £208 per adult, with free kids places.
Finally, with the cheap cost of hotels, you might be better off extending your trip a it and spend an extra night at the hotel. For your party of 2 adults & 2 kids, you could currently fly to Barcelona with Easyjet, going on the Thursday before the GP & coming back on the Tuesday Morning for £343.