I haven't been to Italy in September, so I can't tell you about reservations. I don't think 3 weeks is enough to see everything there is to see without killing yourself. We've been 3 times- we just keep promising ourselves to go back and see something else the next time! My husband loves Venice (well, so do I, but not the same way) I love Florence. There are lots of towns in Tuscany which are lovely to visit and wander around. Rome is great too, but it's such a "big city" that it doesn't feel the same. It's also really easy to get "museumed-out" in Rome. If you've seen one bust with marble stuck to it, you've seen them all.

Ostia Antica is kind of neat, it's the ancient seaport for Rome, you can get a one-day tour if you want. We also liked the Amalfi coast. Pompeii was worth a day, if you're into that kind of thing, just watch out for the people from the cruise ships. They're everywhere, and they think they own the place. And there are SO MANY of them at a time...
I didn't like Naples at all, but they were having a garbage strike when we were there, and it was beyond description as far as trash goes. Also didn't like the train station there and the people trying to make a buck off you even when you didn't want it. Unfortunately, you can't really avoid it if you want to go to Amalfi coast or Pompeii and don't drive.
You'll have to decide what you really want to see and plan accordingly. We prefer to base ourselves for 3 to 4 days in one place and make trips from there. Get familiar with the train system. We usually buy passes good for so many trips in so many days, and use them to get around, but you can also buy single trip tickets for the day trips.
Don't spend
all your time in churches and museums. Do try the local food specialties. I'd rather spend a morning in a market than a museum, but I've been to a lot of museums over three trips. And a lot of markets.

I always wish I had pots and pans to cook with when I"m there... I wouldn't bother with some of the ones the guide books say you "have" to do. I hated the Uffizzi in Florence (while it has some really good stuff, they also have a lot of, well,
stuff. Indiscriminate.) but I love the Opera del Duomo (where they put all the stuff they took out of the Duomo) I love the view from the top of the Duomo and from the top of St. Peter's, and I'm afraid of heights. It was worth it anyway. Siena's cathedral has the stripiest interior ever, and Lucca is pretty all over. I love the wretched excess of St. Peter's and the Vatican museums. Venice is, in itself, one big museum, really. The Amalfi coast is one beautiful view after another.
No matter where we've been, we've found the people to be terrific, the food wonderful, and the country to be lovely. I doubt I'll ever get tired of it. Have a good time!