Best region in Italy to visit late June to early July

There is no right or wrong answer. It really depends on your interests. Just know that July is scorching hot in Italy and that is why Italians escape the cities and head to the seaside or to the mountains in summer. My personal preference would be to go to any seaside location, it doesn’t matter where, or even the mountain areas in the north (Alps) or the Apennines. If I can avoid cities in the valleys, like the ones you mentioned, I would. But if those cities interest you and you rely on public transportation, then go ahead.

Sorrento/ Amalfi Coast area will be packed, and you'll probably have problems with lodging availability, in addition to the infrastructure issues.
I do know folks here like Salerno, and there's good train transportation from Rome. But- you still have the heat.
Have a great trip and fun figuring out your itinerary.

The south will be hot. I know it is quite beautiful but it will also be jam packed with tourists, peak tourist season in a very popular destination. I think I would prefer Sorrento, Salerno and Amalfi coast area in spring or fall over summer. I would suggest you head up north. I would consider your idea to visit Milan, Bergamo, Brescia and Bologna, if those are cities that interest you. I went to Brescia last spring and liked it very much for a two night visit. Not on the American tourist radar. Pretty piazzas, churches, a castle, ruins and an archeological museum. You might consider Verona, a lovely city with lots to see, but it too will probably be crowded with tourists. Not very far from Verona is the pretty town of Mantova. You could head up north to Trento and from there visit the northernmost towns on Lago di Garda. Riva del Garda, Limone sul Garda and Malcesine are all so picturesque. I have been to Trento twice. In Trento you get a combination of Austrian and Italian influence in architecture and food. It is not overloaded with tourists, has a pretty and compact historical center, is surrounded by beautiful mountain scenery, and makes a good base to explore that scenic area with good train connections.

Be aware that Bolzano, while a very attractive city close to the Dolomites (bus service available), is itself positioned down in a valley and can be quite hot in the summer. Be sure your lodgings are air conditioned, and read reviews to ascertain the quality of the a/c. When I visited the area in 2015, a lot of pretty nice hotels didn't have air conditioning. You need to go up into the mountains to be reasonably certain of avoiding high heat, though you could obviously be lucky with cooler-than-average temperatures anywhere.

You might consider Sardinia. It's very rural and the driving can be twisty through the hills, but it has a lot of beaches and the food is amazing, even by Italian standards.

Umbria itself and Tuscany or even Le Marche have so much to offer that I would stay where you are. Cities of the north can be very hot that time of the year.

Thanks so much to everyone who took the time to respond. Now I have lots of other places to research!