north east area of Italy

How long is this trip? The only place you might need to rent a car, would be the Lake Garda and Dolomites areas.

Not knowing which side of Lake Garda,

MXP to Milan Centrale using the Malpensa Express train.

Train. Milan Centrale- Peschiera del Garda. 3N.

Train. Peschiera del Garda- Verona. 2-3N.

Verona-Malcesine (Lake Garda) car rental. 2N.

Malcesine-Bolzano-Ortisei (Val Gardena) car. 4-5N.

Ortisei-Riva del Garda car. 2N

Riva- Torino Drop car. 3-4N

Train. Torino- Trieste. 3-4N

To cover all of these areas, you would need at least 3 weeks.

For an estimate of train and drive times use rome2rio.com.

To make it even more of a challenge, Bergamo has a beautiful medieval district on top of a hill, and it's an easy train ride from Milan.

This sounds like a wonderful trip! I second the recommendation to visit Bergamo, very much worth a visit, especially because it showcases a side of Italy that is so different from Tuscany or "southern" cities like Rome or Naples!

Take the train from Milan to Verona where you can rent a car to visit the Lake Garda area, Trento, Bolzano, Ortisei, Val Gardena. Then take the Great Road of the Dolomites to Cortina d’Ampezzo. Head south to Padua. Ditch car on arrival in Padua (ez train to Venice) and take train to Trieste. I made that trip in reverse just over a year ago and it was wonderful.

Love the recommendation acraven gave for Bergamo. We visited a few years as a day trip, wished we had more time. Great place to get over jet lag. Try this instead of the itinerary I posted above.

Milan-Bergamo 2 N

Bergamo-Verona 3 N visit Peschiera del Garda from here. Train about 15 minutes. Pick up car as you leave Verona.

Verona-Ortisei 4 N. visit Malcesine on the way.

Ortisei- either Trento or Trieste (drop car) you really don't have time for both. 4 N.
Visit Riva del Garda on the way.

This is fast paced but can easily be done since your locations are relatively close together.

So do I book flights to Milan and back. Or is it better to fly home from another city.

Lago di Garda and the Dolomites are best with a car. You would end at Trieste and could drop the car there, or keep it to drive to your departure airport.

Venice would have the closest major airport to Trieste, but look carefully at the flight times. Many flights to connection points for US flights ( Frankfurt, Amsterdam, Paris, London) depart Venice very early. MIlan may offer more options, as well as better prices. The train from Venice to MIlan is under 3 hours.

Consider BLQ for your return flight.
We chose BLQ for our arrival airport over VCE due to convenience of flights.

Consider Treviso as a place to stay.
We loved it.
https://www.nytimes.com/2019/06/10/travel/venice-treviso-overtourism.html