How much time do you have? I don't necessarily agree with staying in one place for a week. We move much more quickly than that. If you have as long as 2 weeks you can see almost all of Italy. Here are a few suggestions:
The Amalfi Coast is a must see. I love it. Rome is great, while Florence has more art, and is definately worth seeing, it is crowded.
Capri is worth a day trip. Not worth an overnight (unless you don't mind paying lots of money for a room or you want to shop (barf)).
Think about getting off the beaten track. Rick Steves is pretty good for this. Think about Southern Italy...no tourists, much cheaper, very interesting stuff...like Matera. (We love this hotel:
http://www.hotels.com/property.do?mtnHotelID=330351&js=1&zz=1174356216171 you stay in "cave" rooms) Not far from these:
http://www.theculturedtraveler.com/Museums/Archives/Trulli.htm also worth seeing. and only 3 hours from Naples.
If you are going to rent a car (I recommend as you can cover more ground) use a route planner online like this one:
http://www.viamichelin.com/viamichelin/int/tpl/hme/MaHomePage.htm It will tell you distances between places and importantly how much time it takes to drive from one place to another (invaluable in Ireland where there are almost no freeways! and in parts of Italy like Amalifi coast.) Things are closer than you might think. Florence to Naples is only 4.5 hour drive.
No city has enough to really need more than 3 days...and that's a big city. Cover some ground.
Go to the library, check out picture books of the area and try to get some travel videos (Steves or Rudy Maxa are both good). Watch and look and if something interests you, go there. We love the hill towns around Florence (San Gimignano is one of favorites, as is Sienna but many others) and the Cinque Tera. Umbria is also great.
Try to plan your trip, if you do decide to cover more ground, so you don't have any stays of only one night in a hotel. Tires you out too much. Arive late, stay 2 nights, leave early. Pack light! There are more stairs than you might imagine and not all that many elevators. You'll be lifting that big suitcase up stairs...better if its light. You can wash stuff out as you go. Last time we did Italy I didn't anticipate the heat (it was March, I didn't expect it to be that hot) I bought 2 cotton dresses and washed them out in the room. Wore nothing else (well, undies) the whole trip.
Get up early in the morn, hit the first sight when it first opens, tour groups take a lunch between 12 and 2 so visit sights then when there are fewer tourists and eat lunch later.
Try to stay in smaller independently owned hotels, more interesting. Steves has lots of budget recommendations.