3 Week Trip to Italy in July for Family of 4

•Fly in to Rome (4 nights)
•Train to Naples for Amalfi Coast (5 nights)
•Fly to Milan for Lake Como (3 nights)
•Train to Venice (3 nights)
•Train to Florence for Tuscany (5 nights)
•Train to Rome (1 night)
•Fly out of Rome

A car would be a good idea perhaps for the last portion, Florence to Rome Airport.

When you say the mountains, are you meaning the Dolomites
North of Venice, or the mountains north of lake Como or into Switzerland, the alps?

Having a car in Tuscany will give you a lot more flexibility to better explore hill towns than with just train tickets. I can highly recommend Siena, Montepulciano, Pienza, and Montalcino but see Rick's Italy book for a longer list. With a car, you could also stay in an agriturisimo which is a fun experience, although with four of you I'd probably encourage you to rent an apartment for savings. (Great ones on Airbnb all over!)

That you so much for the replies so far.
By mountains I mean Lake Como area.

not sure how long you will be flying/from where? But I would see Rome on the way out, it is exhausting in the heat and you will want to be on the time zone firmly when there.

The only way I see Amalfi Coast and Lakes fitting in, even in 3 weeks is going to Naples directly from Rome upon arrival, high speed train 1 hour from Garibaldi train station, check this forum for more information about a new direct bus service from Roma/FCO to Naples train station!

Or take private car from Naples to Positano/Amalfi, best way after all the travel or with 4 ppl you can splurge on Rome- Positano car service.

Positano etc 3-4 nights/ make it 5 if going on day trips to Capri or Paestum
Head to Naples train station
Then make your way to Venice by train 3 nights
Lake Como area towns 3 -4 nights (Milan is easily skipped, ) but you could stop for a few hours on way through to take train to Florence / Tusacn towns
Florence 2 -3 nights. 2 if you aren't into museums, 3 -5 nights Tuscan towns
Rome 3 nights, to see main highlights, 4 if going to The Vatican or Ostia Antica ( ruins similar to Pompeii)

IMHO doing Lake Como area and Amalfi Coast is a lot in 3 weeks, I would pick one of those two.
My general philosophy is stay below Florence or Rome upward....six places, in 21 days is a lot of travel, I would see Como on a trip to Switzerland/ Germany.

It is likely to be extremely hot in July, and a bit worse in the south than the north. Keep that in mind as you decide whether to include the Amalfi area and/or Lake Como. Based on what I experienced in 2015, I'd seriously consider substituting some time up in the Dolomites for one of the hotter destinations. Three weks is a long time if you get caught up in a heat wave.

Thank you all for the great suggestions! Now I feel I can come up with an itinerary that will best suit our family and timing......and decide what gets set aside for the NEXT trip at a different time of year.

Just off the top of my head, I'd (numbers in nights):

Arrive Rome
(Train Rome/Florence)
Florence 3
(Rent car--watch out for ZTL's!)
Tuscany 5 (San Gimignano, Montepulciano, etc.)
(Return car)
(Train Florence/Verona)
Verona 3 (visit Venice, Lake Garda)
(Train Verona/Como)
Lake Como 3
(Train Como/Rome)
Rome 7

This trip skews a little north because July is hot in Italy. Florence can be like an oven but you have to start somewhere, and you can pick up the car outside that town, and really enjoy knocking around Tuscany. From there, I would originally travel northeast to Bolzano and Bressanone in the Dolomites but it's a long way from Como, so you'd have to pick either or IMO. Skip Milan, daytrip it to Venice from very nice Verona, and concentrate on Rome at the end, maybe getting an apartment if there is something still available. The Eternal City is to be savored.

Enjoy your planning!

My family of four will be in Italy for four weeks this July. We will be doing a mix of car rental (for two weeks) and trains. This will be my third time driving in Italy (I also have trained exclusively through Italy, too.) In my opinion, driving is more economical for a family of four, more relaxing (you can stop when and where you want/need to) and it offers a bit more flexibility, which is great when you happen upon a place you want to visit that wasn't on your radar. Your thought of renting after Rome is smart. Don't have a car in Rome. It is the only city in the world where I will not drive--too crazy! But the rest of Italy is super easy to drive around. If you do decide to rent a car, take a bit to review the Via Michelin site. It breaks down 'costs' for driving in Italy (ie: tolls, gas prices, etc.):
https://www.viamichelin.com/
Also, someone mentioned that it will be cooler up by the Dolomites. We will be visiting friends in Bolzano and apparently it is hotter there in July than Florence. Just something to keep in mind if you're trying to avoid the heat (though that's virtually impossible in July.)
As for itinerary, I think Jay offered a good suggestion if you want to hit the 'highlights'. Good luck with your planning and have a fun trip!

Bolzano is down in a valley and can indeed be very hot in the summer. To make it worse, a lot of the city's hoteliers are in denial about current weather patterns and have not installed air conditioning. But Bolzano is not in the Dolomites, just very close to them. To be sure of escaping the heat, one needs to stay at altitude.