13 nights in Italy - flying in & out of Milan - itinerary advice for our family?

This sounds like a good plan to me. Would you plan a day trip to Pompeii from Rome? It would be a long day. You may need to consider the stamina levels of the various family members. Walking in Pompeii is not all smooth. Will your (not old) dad be ok with uneven cobblestones etc? Have a great time!

Arrive MXP.
Train from MXP Airport to Venice (via Milano Centrale, where you will change trains)
Venice 3 nights
(train all the way to Sorrento, via Napoli Centrale)
Sorrento 3 nights (visit Pompeii on the way in/out or while in Sorrento)
Rome 3 nights
Florence 3 nights
Milan 1 night.
(total 13)
In the above itinerary I chose to go to the farthest destination after Venice, then work your way back north to Milan. I wouldn't try to go to the farthest destination on the day of landing. That would be too much on the day of arrival. Also Venice is a good place to be while you get used to the time difference. No cars that can run you over while you half asleep. At most you can fall into a canal, which will definitely wake you up.

You don't need a car for this itinerary, just trains
www.trenitalia.com
Station names must be spelled in Italian:
Malpensa Aeroporto
Milano Centrale
Venezia Santa Lucia
Firenze Santa Maria Novella
Roma Termini
Napoli Centrale
Sorrento is served by a private commuter train called Circumvesuviana, not by Trenitalia.
www.eavsrl.it

If you want you can also visit Pompeii from Rome on a day trip (still via Naples).
In that case allocate the Sorrento nights to Rome (and maybe Florence)

For a day trip to Pompeii from Rome consider a private hire. We are 6 people going to Rome in February. Milesandmiles.net, recommended by Rick, has a day trip, door to door, for 775€. Their website says Pompeii and another place, can't remember. Anyway, they will customize for you. We are looking for Pompeii and Sorrento. It is 11.5 hours, starting at 7:30. It might be easier for you then taking the train, transferring, etc with the kids and dad in tow. Also, the kids/dad can take a nap either way to help with the long day. I know it sounds expensive, but we have found after many trips that sometimes you have to splurge. Good luck with the planning.

You may disagree, but since Rome and Pompeii are your top priorities I would spend more time there and forsake Venice. You can always return to Italy with a focus on the north/northeast some other time. Here's my suggestion:

Day 1 Milan (either now or at the end of trip, see Last Supper and Cathedral)
Day 2 train to Rome
Next few days, see sights in Rome
Train to Pompeii or Sorrento, stay there at least 2 nights (touring the buried city of Pompeii is a full-day affair)
You could also see Herculaneum and/or Oplontis and/or Paestum
Return to Rome or go straight through to Florence
Next few days, see sights in Florence, possible bus trip to San Gemignano, Siena, or other Tuscan hill town
Return to Milan

Hope this is helpful!

Thanks to all for your helpful ideas. I do think we can't miss Venice. There's just no place else like it and the kids are dying to see it. We will budget enough time for the Duomo and the Last Supper in Milan on our way back...that's all I really need to see there on this trip.

I think my dad will be ok in Pompeii. It's hard to say. He wants to go. He's pretty self sufficient, though, and can sit somewhere and read if we need to park him someplace.

I love the idea of the Pompeii day trip from door to door...but ouch. That's a lot of money! Does anyone have ideas for less expensive options?

Pompeii vs Herculaneum - what are peoples' thoughts on that?

One more wrinkle...the reason I had planned to go to Florence first and then to Rome was to miss Holy Week in Rome, which I was afraid might be a zoo. Am I wrong?

Milesandmiles.net, recommended by Rick, has a day trip, door to door, for 775€.

Holy cow!...I would keep in mind that, although 11-12 hours sounds like a long worthwhile day trip, these are not 11-12 hours "on the ground". At least 6 hours will be taken up just for the round-trip drive, so your actual "touring time" is only 5-6 hours which is really not a lot (that's why Rome does not make a good base for seeing Pompeii). Only 2 hours on this tour are dedicated to actually seeing Pompeii, the rest is other towns like Positano. That's a really poor deal, I'm sorry to say, if your priority is to "see" Pompeii. Two hours is way too rushed.

Better to stay close by (Sorrento etc.) overnight for a few days and use that city as the base from which to go on your own and stay as long as you like (I stayed more than half a day and took the Circumvesuviana from Sorrento, very inexpensive). I would not do Pompeii as a day trip from Rome - bad use of time and really, really long day. I have never seen a day tour cost as much as this one but that's because you're paying a whole heck of a lot just for the private transport that could easily be done via train. Unfortunately, I did not make it to Herculeum (Ercolono) but really regret it. It will be much less crowded than Pompeii.

The volcanic ruins of Pompeii comprise a big city. You can walk all day and not run out of interesting things to see. Herculaneum (Ercolaneo) is much smaller but also fascinating. It is right on the coast whereas Pompeii is inland.

The full fare standard ticket on a high speed Frecciarossa from Rome to Naples is €44 each way. Children 4-12 pay 1/2 price.
The Circumvesuviana commuter costs next to nothing.
So 3 adults and 3 children would pay about €400 round trip by train.
That is the full fare. If you purchase in advance you can save over 50% on the adult fare, and often children travel free with an advance purchase with the "Bimbo gratis" special offers. The only draw back of discounted tickets purchased in advance is that once booked you get no refund in case of cancellations or change of plans. Those tickets cannot be changed or reimbursed once purchased. You are locked on that choice.
See the trenitalia website for details.
If you just want to visit Pompei, a day trip is doable by train. You can be there in less than 2 hours and there are at least 2 trains per hour from Rome to Naples (in each direction). 3-4 hours in Pompei are sufficient. If you want to add Positano and the Amalfi coast, then staying a few nights in Sorrento is a better option.

It's hard to explain why walking is so difficult in parts of Pompeii, but I'll try. The worse part I recall (and right in the heart of the town) had extremely narrow sidewalks. The streets are paved with large, oddly shaped stones, had filler around them when they were originally paved. That filler has long since disappeared, so you have to walk on uneven stones that are not flat on top, with deep ruts between them. It's hard for anyone, quite challenging for someone who does not have good balance.

Herculaneum is a much smaller site, but gives you a good idea of what an ancient Roman city was like, including restaurants, public baths and toilets.

I'm thinking the same as Agnes that Pompeii is going to be a really long day from Rome, and especially tiring for the eldest and youngest of your group. That one is much easier to do by inexpensive commuter train from Sorrento so if you can work it into the schedule, I'd go there for a couple of days.

I'll throw this out as an alternative: take a look at Ostia Antica? This one is a much, much shorter distance by (cheap) train from Rome, and will be much less expensive/time-consuming for your mutli-generational group. You can probably do it in, oh, 1/2 day, and it sees very good reviews.

Official Site:
http://archeoroma.beniculturali.it/en/archaeological-site/ancient-ostia

Rick's page on the scavi:
https://www.ricksteves.com/watch-read-listen/read/articles/ostia-antica-near-rome