Two months in north Italy - where to stay?

You mentioned Verona, and my first thought was "somewhere around Verona". Verona itself is too big for your specifications, and still not quite in the mountains, but Rovereto a little bit further up the Brenner highway would be a possibility - it has excellent train connections. Or perhaps Trento, but I am not sure it is that much smaller than Verona.
Bassano del Grappa also comes to mind, but the railway connections are worse.

In a very different part of the country, Cuneo would be nice too, especially since I believe the road to Nice (Tende tunnel) will be open again by next year.

How many locations are you considering to split your 2 months' stay?

There are many places that would have many amenities you want (safety, vibrancy, shopping, apartments with wifi, parking, culture and history, cozy city center, access to train/road connections, etc.) but to be able to access Liguria, the Alps, Amalfi Coast, Tuscany, SE France, and Switzerland from one place, you need an airport and a private jet.

Base case, we plan to stay the whole time in one place, but then take some multi-day trips to other places.

Balso, thanks for the suggestions. All three look interesting at fist sight.

Does Padua work? It has good rail connections and a cozy downtown or Mantua (although I don't know about rail there)? They are in in Liguria but would offer to you other things on your list.

Bergamo.

Plenty of destinations throughout Europe are also reachable from Bergamo airport (BGY), Italy's 3rd busiest airport.

My family really enjoyed our stay in Aosta, where we spent 5 days. It has a nice walkable area, great playgrounds, mountains close by, great hiking (and mountain biking if that is your thing), and is close to Gran Paradiso National Park. We really enjoyed the area. We rented a car to explore the area. Good luck with your search!

I'd suggest doing some route-checking on the ViaMichelin website for potential driving destinations and the Trenitalia website for rail destination. I think you're going to find you'll be paying double lodging expenses quite often if you opt to stay in one place for the entire time.

ViaMichelin website
Trenitalia website

I agree with acraven. It's not enough to look at a map and think a place looks central. Make a list of the places you really want to see and try out the transport connections.

Also, how long do you plan to spend in Tuscany and Naples/Amalfi? And on the same trip, or separately? That could mean a lot of travel back and forth.

It's tempting to think that 2 months is a long time, but if you're planning to visit SE France, Switzerland, Tuscany, Naples, Amalfi, the Alps and presumably the area where you're basing, plus cope with work and three year old, that sounds a lot to me. Make a trial itinerary, about what you would do each day.

I would choose 2 or maybe 3 bases. Three weeks in each place is long enough to get the feel of a place, and will save you a lot of time and stress travelling, and paying double accommodation costs.

Some have pointed out that our long wish list of places to see would include lots of travelling, and suggest that we stay in two or three places.

This is well pointed out; however the wish list is a wish list and we are fully aware that we will not be able to do all of it. (E.g., we have been to Tuscany before, and may skip it, and France is very much a nice-to-have). Relocating might be an option, but to save the hassle of moving, we will do so only once, if we do.

Padua is IMHO too far from the mountains, in the middle of the Po plain.
Bergamo seems to be an interesting option. Short distance to Milan, the lakes and the Alps. Any suggestions to where in or near the city?

We recently spent a week in Roverato and it was fantastic. If you want to travel up and down the Brenner/Verona line it is a great base. Trento would be a great choice as well, but it is bigger and didn't feel as "of the mountains" as Roverato. Obviously a totally subjective opinion.

Have you looked at the Aosta Valley? We just returned from it and it is also amazing. Look at the villages between Aosta and Ivrea. We actually thought Aosta itself was a little overdeveloped and touristy, but the other stops along that line were great. The whole line connects to Turin when you are ready to return to the rest of Italy!

Bergamo's Citta Alta is the city's tourist highlight, but the train station is down below, which would mean extra time for any out-of-town trip.

Bergamo has little express-train service. Most trips will start with a Trenord regional train. I think you'll be surprised about how limited your options will be for quick day trips. For example, Verona's at least 84 minutes away (that's on the rare 8 AM Frecciarossa; most trains take over 30 minutes longer). Getting to Bolzano would take almost 4 hours; even Rovereto is 3 hr. 1 min. away.

acraven,

Thanks for info on Bergamo. So far, I think Rovereto is the runner-up, with short distance to the Alps and the lakes.