Almost 3 weeks in Italy--family with 10 year and 14 year old that are great travelers and foodies

I think that sounds like a great trip. In Puglia, stay in a masseria (it's the Puglian version of an agriturismo) around Fasano or Ostuni, preferably with a pool. You'll be well-located for all kinds of day trips.

Bologna might be bigger than you expect. You might want to pick somewhere smaller for a lunch stop.

From Rome, if you don't want the hassle of flying, you could take a train (4-5 hours) to Bari and pick up your car there. It's just a little north of Brindisi. You could drive to Matera from Bari (about an hour), spend the night, then head for your masseria or hotel the next day.

If you'd rather stay in a small town near the sea than in a masseria, look at Polignano al mer (sp?), or Torre Canne or Gallipoli.

Day trips -- there's a marine reserve north of Brindisi and the trulli villages (Alberobello et al) are a must see.

What a great trip this will be!

I'm not sure why you would want to rent a car until you get to Puglia. From Venice you can take the train to Florence (where I would definitely spend more than 1 night) with a stop in Bologna for lunch. Instead or in addition to Siena, stop in Orvieto for a couple of nights before Rome. Then on to Puglia. The bus from Siena to Rome actually takes less time than the train.
My thoughts:
Venice 3 nights
Train to Florence 3 or 4 nights
Bus to Siena or Orvieto 2 nights
Rome 4 nights
then Puglia for remaining time.
You don't mention where you are flying into and out of. Might make a difference in recommendations.

I was thinking the same thing. I don't see the advantage to driving from Venice to Tuscany. If you want to tootle around with a car in Tuscany, then it may be better to stay in an agriturismo than in any of the towns (like Siena and Florence) that have historic centers with ZTL no-driving zones. You could stay in Siena and/or Florence carless and travel by train or bus to visit other towns, and avoid the danger of the stiff fines (very easy to miss the no-entry signs) and steep parking prices, not to mention the hassle of trying to find parking!

I had dinner last night with a local couple in Salerno and her eyes lit up when she talked about the weekend they spent in Matera.

I agree with the points the others have made but also want to point out that the mid-summer heat in Puglia is positively wilting. With a car you will at least not be standing around outside waiting on buses, but it will still be enervating. I know stops at beaches will help, but a week is a long time.

I understand that you are stuck with July, but I do wonder whether you wouldn't be happier spending your rural time in the Dolomites rather than in what I suspect may be the hottest part of Italy. At altitude in the Dolomites, you would be assured of moderate temperatures.

I hope you are from the southeastern US or Texas. I grew up in Florida and North Carolina. I survived July in Puglia, but it was a close call. I can't imagine what it would be like for a family from somewhere like the Pacific Coast.

Thank you everyone for your replies and advice. A lot of great pointers! It's funny you ask if we are from the Pacific Coast, we are actually from California! But from Paso Robles area where regular summer temperatures are 90+ degrees and we love the heat. However, I imagine it is more humid in Italy. We want to spend the last 4 days at least at the beach areas, relaxing, after touring around for the mornings.

We are flying into Venice and were going to fly out of Puglia, but the flights were too uncooperative, so I just bought tickets flying out of Rome. That might change up our order of places to visit. The flight out of Rome is early morning, so it would be nice to spend at least 2 nights in Rome before we fly out. I would hate to be on a train one day and then fly out the next.

I like the idea of using trains and buses more. We were going to stay at a friend's parents' agriturismo outside of Siena. I will ask them if they suggest having a car.

I would appreciate any more thoughts you have. Thanks.