I was surprised to find I was not a fan of Milan or Venice... but glad I went, if only to see the bellringers at Piazza San Marco and Da Vinci's Last Supper.
To me, Florence is a most magical place. I do have lovely personal memories there; I just always found it to be less overwhelming than Rome, and again, just felt very very special. If you go, you must visit the flea market, and rub the pig's snout - there's a brass pig, and legend is, if you rub the snout, you shall return one day!
Plus, you've got the Uffizi and the Ponte Vecchio.... and David... and....
Secondly, I'd recommend the Amalfi Drive highly... gorgeous, lovely. I mean, I've not seen a part of Italy that's not gorgeous and lovely, but again, found Ravello, a very small village at the top of the hill, another magical place; bit o' heaven on earth.
YES - stay away from Harry's Bar! I remember not even seeing any windows to look out on anything there, just crowds of tourists
YES - always stray slightly off the main streets and look for the out of the way places to eat.
If you're a fan of ancient history, Pompeii is very near to Ravello, and fascinating.
And, if you do go to Rome (which to me is a given, particularly if you've not been there before), among the millions of things to see and do, another personal favorite of mine is a visit to the Spanish Steps. That was one of the first places I'd ever been to when I first traveled abroad with my family, when I was nine; we stayed at a lovely little B&B right at the top corner, and would walk down every day and have a meal at a little cafe at the bottom. BEST minestrone I've ever had!
*sigh*
I'm going to have to watch Under the Tuscan Sun now...