Rome, Florence and Venice--and take the full 10 days if at all
The "Holy Trinity" of the three cities above is a good place for newbies to Italy to start. Trains are fast and direct between them, and they're well used to hosting tourists from all over the world. NONE of them are going to be "laid back" at all but cities rarely are. I would advise NOT going during the high-season summer months when the heat and the crowds can be a bit much. Spring and fall can be a little more comfortable, although still pretty busy. Allow for at least 10 days as 7 would be a real hustle.
If your 10 days (it's actually best to count number of NIGHTS you have) includes your flights to and from Italy, your itinerary might be:
Day 1: fly to Venice
4: Early train to Florence
6.Florence- train to Rome
10: Fly home
Keep in mind that every time you make a change of location you're going to lose 1/2 a day in the process. I wouldn't expect to be able to do any sort of "current cultural" immersion in this amount of time, and please don't bypass ALL of the major landmarks/museums as they're very important to the proud history and culture of Romans, Florentines and Venetians. Italy, as a unified country, is younger than the United States! The original purposes of many of those landmarks wasn't for tourism, and locals/Italians from other parts of the country visit them too.
You don't have to, say, do the Uffizi in Florence if art isn't your thing but I wouldn't skip some of churches where vestiges of the birth of the Italian Renaissance are still beautifully evident 600 years later.
Rome is not Florence, is not Venice: they're all different, and fascinating wanders can be had in all!
Editing to add: you mentioned that you had a "wonderful Rick Steve's book vacation in Paris and Prague"? Then a good place to start planning might be to pick up his book on Italy or books on Rome, Florence and Venice. Spend some time with them to make a personal "must see" list.