In northern Italy in summer is usually very warm and humid, so some days among the mountains are a good suggestion. Valle d'Aosta is wonderful. You can base yourself in Aosta or in a village, but is easier move by car and hike. The same is true in the Dolomite, but is a very wide area so you can decide where to stay. As lakes Maggiore, Como's and Garda's are all wonderful and plenty of activities.
Venice as base is not one of the best, but spend there some days worth the travel. Better base to explore Veneto is Padua.
Milan maybe can be kept as base for the first day after arrival, or before leave Italy by plane: there are some wonderful museums there.
For Emilia-Romagna Bologna is a perfect base, because is an important train hub, so you can reach all cities without trouble.
For Piedmont you can decide to base yourself in Turin, but even in a town in the Langhe (the southern part of the region): the second one is the most known area of Piedmont by foreign tourists.
Liguria is very nice and the Cinque Terre are delicious, but usually in summer are overcrowded. You can spend there some days, but are not probably a good hub to explore. Is much bette Genoa.
For beaches, if you need some days to rest and relax, I can suggest northern Tuscany (Versilia) or Riviera Romagnola. Another good area could be the Conero, near Ancona.
Tuscany is very well known by Americans. As base you can choose Florence. For Umbria you can choose Perugia.
In general I can say that the public transport in italy is very good to move between cities and main towns, but is often weak to reach villages or natural areas. There is a good bus system, but designed mainly for students and workers, so often the timetable don't fit very well a tourist needing. For example a lot of buses between 7/8:30am and a lot between 5/7pm, but only few in the middle, so you can reach a village you like to visit, but you have to spend the whole day there. Moving by car is much easier, because you can visit different places on the same day without waste a lot of time, or stay longer if you a like the place or leave before if it disappoints you.
In the tourist areas there are even daily group tours, so you can join one to visit an area.