If you're only interested in seeing the Colosseum (and presumably the other Ancient Rome sights nearby), the Vatican and associated museums (not sure what that description means), you could probably see that short list in 3 days or less. But if you're saying you have 3-4 weeks in the country, I hope you'll give more time to Rome. We've spent about a month in Rome over several trips and still haven't seen everything. I would stay no less than a week on a first visit.

You want to see Pompeii, but do you not have any interest in Naples, Capri and the Amalfi Coast? You could see Pompeii in one long day trip by train from Rome, but there's much more to see than just Pompeii. For a start, the archeological treasures removed from Pompeii are now housed in the Naples archeology museum. You could easily spend 5-7 days in this area.

Tuscany v. Piedmont is personal preference and, to a lesser extent, how late in "the fall" your trip will be. I love both areas but prefer Tuscany (and Umbria) over Piedmont. We travel most often in late October and have had great luck with weather at that time in all three regions, although Piedmont was, not surprisingly, colder. You'll need a car which ever area you choose.

Will you need to return to Rome to fly home, or can you depart from Milan or Pisa?