First time in Europe is always a challenge in choosing among many options. As you have two young children, avoid hustle and make things easier is important.
A) city hopping: use air and train, dont drive. You can easily cover London, Paris, Rome, Milan and maybe Vienna Salzburg in 3 weeks. You only need to plan for regional flights (try easyjet.com, ryanair.com) and city hotels (try booking.com), and within cities use public transit or taxi. You may want to book your intenational flight US-London and Rome-US to save time.
B) Country exploring: you can rent a car after visiting London and Paris (5 days), start driving south of France along one of the two routes (west via Bordeaux to the Province -May is the best season there - and Cote d'Azur, or east via Alps town Chamonix and Geneva Laussane) to Italy (7 days). Then drive to Tuscany via Milan, Florence, stay in a Villa in Tuscan towns (recommend Chianti region) for 3 days, finising at Rome (3-4 days including Vantican). If you have to return car in Paris, you may have to bypass Rome and drive north via one of the routes above in France back. This may take additional time though, but I dont know if you can pick up car in Paris and return in Rome, maybe possible.
