We did what I call a "if it is Wednesday, we must be in Florence" tour of Italy, because of limited time. Most travelers would consider it crazy rushed.
Day 1 afternoon arrival /2: Venice St Mark's, Doge's Palace, DFS mall (great canal view), glass blowing demo. Depart for Florence late evening.
Day 3: Florence: Michaelangelo's David, Duomo, Uffizi Gallery, late evening Pisa/Leaning Tower. Overnight in Pisa.
Day 4: Cinque Terre (express train from Pisa only an hour), return to Florence late evening, then Piazzale Michaelangelo in Florence
Day 5: Depart for Rome. Vatican/Sistine Chapel official tour, and also Coliseum
Day 6: Trevi fountain, Spanish steps, just walking around, late evening depart for Athens
We researched trains and bought tickets in advance for the shortest transit times between Venice/Florence/CT/Rome. And we did find time in Florence for my wife to shop for leather handbags, and in Rome for other shopping. In a sense, our snapshot tour could be similar to conducted tours, except that we covered many more places.
Based on this snapshot tour, we will plan a longer trip to Italy in the next year or two. Incidentally, we arrived in Venice after a connection in London (our roundtrip tickets were SF Bay Area -London). Our trip to Europe was 14 days, excluding travel days from/to the US.