Paring down our N. Italy itinerary

Fly into Nice and take the train to "Prettiest town between Monaco and Genoa". There are numerous recommendations.
Spend 4 days to relax and get over jet lag.

Train to Milano.
Buy your Last Supper tickets today.
Spend 4+ days. Big city. Lots to do.
We are planning 6 days there this March. It is our arrival city.

Train to Varenna. 1 hour/every hour. If Bellagio, then ferry from Varenna to Bellagio.
3 days.
For us, this is our "beach town".

Train to Verona (via Milano)
3 nights

Train to Bologna.
5 nights
Many day trips are possible via train. We went to the Ferrari museum in Modena. It was quite enjoyable.
We stopped in Ferrara on our way north. Ferrara has 2 attractions: the castle and the Jewish Museum (which is closed on Mondays).

Train to Treviso. Skip Venezia
We loved Treviso. Small. No crowds. Canals.
https://www.nytimes.com/2019/06/10/travel/venice-treviso-overtourism.html
Can do day trip to Prosecco country to the north.
3 nights.

Thanks, @davebarnes. Good info, especially the train info. Interesting that you suggest 4+ nights in Milano. We had considered no more than 3. You also mentioned taking the train from Nice to the "Prettiest town between Monaco and Genoa"... what is that town?

Addendum my OP, visits to the Domomites and the lake district are very high on our list, so any itinerary will include several days in each. Also, we plan to spend the last 4nights/3days in Venice. We appreciate the advice to the contrary and understand the whole overcrowding/overloved situation in Venice, especially when the cruise ships come in. That's why we want to spend several nights there to experience it early/late and visit the quiet areas. We did a 2-night stay there in 2009 and didn't really get to see much, and we've been wanting to return to spend our middays "getting lost" in the out-of-the-way portions of the city. We also fly out of VCE.

"You also mentioned taking the train from Nice to the "Prettiest town between Monaco and Genoa"... what is that town?"
Surf the InnerTubes®.

Research the Dolomites carefully. I think you are too early for many of the attractions and many places will be closed. We went camping near Corina in late May one year and it was no fun. Look at this thread: https://www.tripadvisor.com/ShowTopic-g194745-i2240-k7038802-Cortina_d_Ampezzo_in_May-Cortina_d_Ampezzo_Province_of_Belluno_Veneto.html
Save it for a fall trip or perhaps you could stay in Bolzano and drive in to some areas.

Thank you for your info and link, @Mike. By the time we'd get to the Dolomites, it would be the second week of June, though that may still be too early to do anything other than drive around an ogle the mountains. Perhaps we'll need to cut down on the number of days there. We're reluctant to remove the area altogether, but your comments are worth considering.

Given the latitude of the lakes district, will it also be also cool up there during the first few days of June to enjoy? We're locked into our flights (arriving LHR on 27 May and departing VCE on 18 Jun), but we can go anywhere during that time. Perhaps we should consider changing focus and visit the eastern coast or the far south instead of the northern part of the country, though that would require a long train ride or short intra-country flight to Venice at the end. Again, we have lots of flexibility.

The interesting thing about this trip is it was originally planned for Croatia & Slovenia, starting in Dubrovnic (which is why we booked through LHR) and ending in Venice. We then decided to save a visit to that area (including B-H, Serbia, and Albania) for a tour, something we rarely do, and changed our focus to Northern Italy instead. In retrospect, perhaps we should have chosen somewhere further south during this time frame.

Decisions, decisions... but a fun problem to have!

I hit the Dolomites in mid-summer so have no advice about that area in May, beyond the fact that it's mostly the altitude of the mountain villages (like Ortisei) that puts you at risk of chilly weather. Bolzano, Bressanone, Chiusa, etc., are down in the valley and should be fine in late May. In fact, I'd worry about excessive heat by June, if not May, and would be certain my lodgings had air conditioning. (Not all do.)

I don't remember precisely what time of year I visited the Italian lakes on my various trips, but I think on the south side of the Alps you'll be fine. However, there's usually some risk of rain, which can bring with it cooler temperatures, in the places not guaranteed to be deathly hot.

Rather than accepting my musings at face value, I suggest that you visit the website timeanddate.com and check out the actual, day-by-day, historical weather statistics for the destinations you're worried about. The data goes back 10 years, and you can see what happened on precisely you dates.

If you decide to include the Bernina Express, do some digging about the best place to stay on the Swiss end. I remember reading that St. Moritz isn't the best choice, being very expensive and not all that attractive. I think Pontresina was recommended instead, but I haven't made that trip and am not sure.

Thanks for the feedback, @acraven! I will check out the info you suggest.

All: Since we may end up retooling our itinerary, I'll ask this as a general question: if you had 3 weeks in Italy, where would you go? (Other than Rome & Florence) I'm open to suggestions of your favorite places or even a suggested itinerary. Thanks!

Here’s a suggestion for s 3-week itinerary that doesn’t go to Florence or Rome.

https://community.ricksteves.com/travel-forum/trip-reports/solo-trip-to-italy-2018

The only reason I didn’t go to Venice on that trip was that my husband said I couldn’t go there without taking him ( we both love it!). and I had my itinerary filled anyway.

We have been to Torino, and enjoyed that city. We stayed near the royal palace, and it was handy to see the sites for a few days. We left feeling like we would enjoy returning to that city.

We have been to the Dolomites but both times were in September, so I can’t give you late Spring advice. It is a lovely area and a nice contrast if it works for you.

Oh my gosh, @Jean... your trip report was fabulous! I would love to be your travel partner someday (I'm certain our spouses would get along swimmingly, too). Thank you for adding such great anecdotes and insights to your journal. I am adding several of these locales to our Italian bucket list.

P.S. I have family in Lake Stevens, WA. Anywhere close to your Seattle location?

@acraven: I looked up some mountain towns in the Dolomites (Santa Cristina Valgardena, Castelrotto, Bolzano) using the timeanddate website you mentioned, and the high temps in early June the past 5 years showed high 70s to mid 90s F. That makes me feel better about travel in the area during this time frame. Thanks again for the tip.

Bolzano is a very pretty city with an important museum (say hello to the Iceman), but it is down in a valley that can be very hot in the summer; that's probably where you found your 90-degree temperature. A further complication is that the amount of heat is apparently a fairly recent phenomenon and a lot of hotels are not air-conditioned. I suffered through that for 4 nights in Bressanone (also in that north-south valley) and then hightailed it to a four-star hotel in Bolzano (not my usual type of lodging, but it had a/c).

For melowgo: I am partial to not-tiny places with sights of their own where I can spend at least four nights and from which I can make day-trips to a variety of destinations. Two such places in northern Italy are Padua and Bologna. You already know about Bologna. I also really liked Padua, and it was not (as of June 2015) swamped with foreign tourists. Ferrara's about as easy to visit from Padua as from Bologna. Other options from Padua are Vicenza and Verona. And of course Venice if there happens to be something you're interested in there for which day-tripping makes sense.

Melrowgo, yes, my Italian bucket list keeps growing, too! I’ve stayed in 20 plus towns in Italy and am returning this year, taking my 40-year old daughter for her first European experience. With this trip, I’ll add Bergamo, Ferrara and Amalfi Coast to my list of places stayed, and of course, returning to favorites that my daughter requested.

We used to live in Mill Creek which is just south of Everett. Lake Stevens is east of Everett. When we first moved to Washington State 30 years ago, we first lived in Snohomish which is next to L. Stevens. There’s a pair of our sunglasses at the bottom of Lake Stevens! Two months ago we moved to the Coeur d’Alene area of Idaho to retire - loving it!

Good info from all. Thanks for your continued feedback! It is hugely helpful. We're currently deciding between staying with Northern Italy and changing to the eastern coast along the Adriatic.

@Jean: yes, I know exactly where all of those WA towns are. Congrats on your new locale! Beautiful area.