Do I need a car in northern Italy

You certainly don’t need or want a car in the cities you plan to visit. The trains go to these places. I’d check guidebooks (starting with Rick Steves Italy) to get more information. For the Tuscan countryside, you can reach many towns by bus. If you do decide on a car, I’d to a rental from a Florence location. With the tram from Florence to the airport, you might want to rent at the airport as it will have less trafic to the countryside routes. Of course driving in Italy is not for everyone.

No. Yes.

Milano - Genova by train. Under 2 hours.
Milano - Varenna (on Lago Como) by train. 1 hour. Ferry for Varenna - Bellagio.
Milano - {Bergamo, Brescia, Verona, Vicenza, Padua, Venezia} by train.
Milano - Trento by train. 2.5 hours.
Milano - Firenze by train. 2 hours.

Tuscany requires a car if you are going to visit small towns.

I would suggest making Bologna a base as there are myriad trains to almost anywhere.
I would suggest staying in Treviso and making day trips to Dolomites and Venezia. I am not a fan of Venezia. Too crowded and too expensive. One visit was enough.

We were quite pleased with all our train travel last September.

Read this about driving in Italy and restrictions involved [ZTLs]; do not leave anything of value in a car when parked:

http://driventoit.blogspot.com/

Driving is appropriate for the countryside, towns and villages not easily served by public transportation; the Dolmites and rural Tuscany would be examples. Because of ZTL issues, driving is not a favored means of traveling between cities connected by trains. You could pick one area to drive, rest by train; or do two separate rentals.

Train operators:

https://www.trenitalia.com/tcom-en

https://www.italotreno.it/en [private competitor with lower fares on some routes]

I would not want to day-trip to Venice from anywhere else. That means seeing the most popular areas of the city under the worst possible conditions. Stay in Venice proper to see her at her best.

Assuming this is in summer . . . we would fly into MIlan and spend 1-2 nights there, then take the train to Varenna on Lake Como. After 3 nights there, return to MIlan by train and change to another train to Bolzano. Pick up a rental car there for the Dolomites. Then drive to Venice, spending 2 nights on the way on Lago Del Garda ( Malcescine) and maybe another in Bassano Del Grappa, which we have enjoyed in the past).

Turn in the rental car at Piazzale Roma in Venice ( mind the speed limit on the cUseway into Venice, as there are three speed cameras and one can get tickets from all three). You could also turn in the car at the airport, but then you have the time and expense of getting from the airport into Venice.

After some time in Venice, take the train to Flor nice and stay in the city without a car. Rent another car for your exploration in Tuscany.

Depending on what you choose for your departure airport, you might return the car there.

We would skip Genoa with this itinerary.

I whole heartily agree with Acraven. Select lodging on the islands of Venice. It’s extra special in the evenings and early mornings before the day-trip travelers arrive.

And to directly answer your question, we have traveled to all of those locations, except haven’t been to Genoa. We did everything by train. The popular bus option is easier if going from Florence to Siena. If you are taking a week in the Tuscany area, then a car makes sense for that region to wander through the country and stay at an agritourismo.

I think Lola has it right. I know it sounds complicated but in the cities no need for a car. Tuscany & the Dolomites having a car is important. Lake Como no car. Boy I'd skip Genoa. No real reason to go in that direction. Bologna is a good base for a few days as is Florence and Bellagio or Varenna on Lake Como (we prefer Bellagio but the train does go to Varenna). acraven is correct - try to avoid Venice during the day. I stayed a couple night in Giudecca and went into Venice after 3pm.

Thanks to all. Very helpful.
Don