Can I assume you are planning on being in Florence through 6/28, then taking your daughter along for the rest of the trip?
Haven't checked flight, hotel or apartment availability, as it is pretty late in the game, but if I were doing it, I would:
6/23: Fly into Milan, probably Malpensa airport. Get over to Milano Centrale train station, then Trenitalia Fresciarossa (high-speed, 1 hr. 40 min)) to Florence Santa Maria Novella Station. Find lodging (preferably apartment) within walking distance to station.
6/24 - 6/28 (4 nights): Optional day trips to Fiesole, Siena, Pisa by train or bus. Plenty to see in Florence as well. Shop at Mercato Centrale for cured meats, cheeses.
6/28 - 7/1 (4 nights): Fresciarossa (about 1 hr. 30 min) to Roma Termini station. Again, apartment/B&B preferable for 5 people, but stay central to be able to walk around easily. Campo de' Fiori & Trastevere my favorite areas in that regard.
7/1 - 7/4 (3 nights): Regional train to Naples, Amalfi or Salerno. Pros & cons to all three. Do not miss Pompei, but do it as early in the day as possible. Amalfi Coast will probably be overrun that time of year, but is beautiful. Investigate Salerno as a great base for day trips to Pompei, Paestum, Naples and the Amalfi Coast. Might have a better chance of an apartment or B&B there.
7/4: Fly Napoil Capodochino airport back to the states. Will probably have to connect in northern Europe for the trip home to the States.
If serious about taking this trip, you need to get on it now. A lot of flights, hotels, apartments, B&B's will already be gone--don't mean to scare you, it's just reality this time of year, even more so with the cheap Euro now. Ordinarily, with more lead time or non-high season, I'd say find the air then work your way in to lodging, trains, etc. Here you're going to have to do everything simultaneously. But I do think that flying into Milan and out of Naples is the most efficient way for what you want to do.
I did see above that you want to do Rome-->Florence-->Naples for whatever reason, which is fine. A little backtracking but not horrible with high-speed trains. In that case, you could still fly into Milan I suppose, but better--obviously--would be to fly into Rome. I'd still fly out of Naples, though. Again, your flight availability choices could be limited with the lack of lead time.
Good luck!