If "everybody" go to some places, there are reasons. It's usually because they are either
1-more beautiful
2-They have more importance in history
So yes I would do the bucket list of Italy which include places like
-The amalfi coast with Amalfi, Ravello, Positano, Sorrento, Capri. Go on top of the Vesuvio nearby, visit the ruins of Pompei and or Herculanum. Naples for a slice of southern italy. All of this could take at least a week. Or you cut with what you feel you prefer when reading.
-Rome to see this unbelievable city. At least 3-4 days
-Umbria and Tuscany for fantastic landscapes, beautiful and historical villages such as Orvieto, Assisi, Perugia in Umbria(a few examples) or Siena, San Gimignano, Montepulciano, Volterra, Pisa, Lucca, Montalcino in Tuscany as well as fantastic food and wine.another week easy. :-)
-Venice, unique city 2-3 nights at least
-Florence 2-3 days
-Cinqueterre for these unique 5 villages, 2-3 days.
These would be the first places that come to mind. Of course, we can say Sicily(which is a whole holiday by itself), the dolomites in the alps, the lake region with como lake, maggiore or Garda.
These would be the first places I would look, then narrow it depending on how much time you have.
So you know, my first trip to Italy I did Amalfi coast 6 nights, Tuscany 6 nights and Rome 3 nights and I still think it was a perfect first trip for my personal tastes. Made me want to go back
I went back 4 other times... :-)