the name of the city is ORTISEI, please correct spelling if you want to look for proper public transportation.
The train from Milan will go to Bolzano (a train change is required at Verona), then from Bolzano you need the bus to Ortisei.
You should decide which lake you want. If you were driving, I'd suggest Lake Garda, which is along the way between Ortisei and Milan. Without a car lake Como is also doable (need to go back to Milan then a train to Varenna). If you decide to visit a lake just before flying out back to the US from Milan Malpensa MXP (or upon arrival) then Lake Maggiore is also doable.
Milan: I'm not a fan of Milan, so I rarely stay over one night, but if you like to stay longer I suggest the area near the Duomo, or just north of the Duomo (Quadrilatero della Moda, or Fashion District). It's also safe. To go to the airport the fastest way from there is by taking the Malpensa Express train from the CADORNA station.
For the type of trip you are envisioning I'd probably rent a car in Milan (maybe at the airport or after you visit the City upon arrival) then visit the Dolomites by car and later drive to a lake (you can also drive the Stelvio pass from the Dolomites to lake Como). From the lake I would go to the airport the day before my flight back, unless you decide to leave visiting Milan for last, in which case you'd return the car to the city (or to nearby Linate airport) on the day you arrive to Milan from the lake (don't have a car while visiting Milan, it would be a waste since it would need to be parked the whole time at the hotel garage, due to traffic restrictions).