Into Rome and out of Venice - 9 nights

If you’re not into art but want one compact museum in Florence, I’ve got a couple suggestions to consider. You pick:

The Bargello features fabulous sculptures in a single room and the entrance courtyard.

For a bite-sized but magnificent display of fresco painting, in a marvelous setting, where the frescoes were meant to be, look no further than the Brancacci Chapel.

You might want to look into a Venetian rowing lesson with RowVenice.org.

I MUST swim in the sea

When in Venice visit Lido, there is a good public beach.

Rome - Campo de Fiore, very near to Piazza Navona.
Florence - You could always stay in Siena, and take the bus into Florence for the day.
Venice - Rialto bridge. Just wander the back streets to get away from the crowds at St. Marks. It is a small island, you can walk the whole thing.

You are missing some key sites in Rome..the colosseum, the area surrounding, Vatican etc. You should not miss these as a 15 year old will certainly find them memorable.
Florence...not an art enthusiast...OK. You should take in at least Accademia. Uffizi will make you crazy with too much art. Go to Sienna as a day trip, you will love it. Drop Bologna...Pisa, OK, but it's a chunk of time for not much...
Venice...sounds good....too bad you don't like art. The Guggenheim is worth visiting..truly. Even non-art lovers will be floored by the work that is displayed...the good stuff, not all the Renaissance Jesus full of arrows and spear cuts, baby Jesus ad nauseum....Polick, Picasso....good stuff that will make you think and that a 15 year old will relate to.
Good luck.

In Florence the Duomo Museum is not large, has fantastic art, and was not crowded at all when we went ( but I’m sure you will have crowds in July )

I don't think you will have much time to visit any other towns from Florence. 3 nights in all 3 cities gives 2 or maybe 2 1/2 days in each. Hardly enough time to get through just a few of the items on your list.

Only 9 nights, I suggest you do 4 in Rome, 3 in Florence and 2 in Venice.

Rome, aren't you going to see St. Peter's Basilica, the Coliseum, Palantine Hill and the Forum? Trajan's Forum, Capitoline Hill Museum?
With only 3 nights in Florence, forget about Bologna, I have been there, it's OK, but pales in comparison to Florence. You can't do Florence well in 3 days (actually less since you have travel) and even less if you take a day and go to Pisa and Lucca. In fact, I would suggest Sienna instead of Pisa and Lucca, since Sienna is closer. Check out the Hotel Balesteri on the Arno in a great location.

Venice, Burano is good, but with only two nights, you might not have time to see it well (aren't you going to St. Mark's Cathedral and the Doges Palace?).

We love: http://loggiato-dei-serviti.florence-hotels-it.com/en/ in Florence.

It is around the corner from the Accademica Museum which has Michelangelo's David. Check online for how to get tickets in advance.

https://www.florence-museum.com/accademia-gallery-tickets.php?gclid=EAIaIQobChMI64PK05mP4gIVh0OGCh2Q_wUsEAAYASAAEgKWC_D_BwE

OR,

I agree that the Uffizi Gallery is total overload, and I agree that the Bargello is excellent, but the one of the finest sculptures in Florence (in my opinion because it is the opposite of "pretty") is Donatello's Penitent Mary Magdalene in the Duomo Museum, which also has the original Baptistry doors and other original pieces from the Cathedral, which is itself pretty empty. https://en.wikipedia.org/wiki/Penitent_Magdalene_(Donatello) https://en.wikipedia.org/wiki/Museo_dell%27Opera_del_Duomo_(Florence)

From the Duomo down to the Ponte Vecchio is a fabulous shopping area, although it may make your daughter's head spin.

Traveling:
https://www.trenitalia.com/en.html

It will be important to learn the Italian spellings of the town Firenze, Venezia, Roma, and the appropriate stations (RS guidebooks) etc to get the right fares.

90 minutes Roma to Firenze.

2 hours Florence to Venice (Firenze ( Tutte Le Stazioni ) to Venezia S. Lucia)

Although staying in Venice is said to be nice (we have avoided it because of the crowds when we would have gone) we stayed in Padua ( 30 or 50 minutes minutes away by regular trains depending on price; 4.50 Euro for slow 14.90 Euro for fast) which is a cool, university dominated town, with great restaurants too.

When we traveled around Europe when we lived there we used the RS guidebooks, and Booking.com (as long as you select 8.5/9.0 or above ratings) for hotels. The Booking.com website is also really good to use to get price ranges around where you want to stay even if you then contact the hotels directly (as RS recommends). July is not too far away for hotel bookings.

In the context of "relaxing" we limited ourselves to an AM activity, nice lunch, a PM activity, and a nice dinner unless something special was going on in the evening. You "get" your table for meals when you go so you can freely linger and enjoy.

Have great time. Google Translate will be your friend. Don't be surprised if your daughter starts thinking of studying abroad.

Thanks so much for the replies everyone, just what I needed! Really appreciate it!

listened to you all and opted for Rome-Florence (or near) and Venice only, save Croatia for next time, thanks for being the voice of common sense :)

So thanks again all for such great feedback. Going through the R.S. Italy book, GREAT! So looking for a little more guidance as I want to get all hotels books this weekend. We have 9 nights, into Rome early (8:25 am arrival) and out of Venice at 1030 am.
What region do you all recommend for hotels in Rome (a coworker suggested Spanish Steps area)?
As far as days in each the book give a pretty broad range (p.12-13)
Should we do: a) 3 nights in each of Rome, Florence and Venice. OR
b) 4 in Rome, 3 in Florence (noting were not art buffs but daughter wants to see Leaning Tower of Pisa) and 2 in Venice OR
c) 4 Rome, 2 Florence, 3 Venice?

R.S guide lists

Thanks again all, I'm sure no matter what we do it will be an amazing experience.

We were in Rome last week before starting RS South of Italy tour. We stayed at Hotel Due Torri, not far from Piazza Navona. It was quaint, quirky, clean, friendly and safe. Easy walking distance to every thing in Old Rome. http://www.hotelduetorriroma.com/en/

3 nights is really only 2.5 days. So if you want to day trip to Pisa- I'd have to say Rome 3, Florence 3, Venice 3.

We like to stay near Pantheon/Piazza Navona. You can walk everywhere from there - it is about as central as you can get.

Spanish Steps area is not all that attractive, lots of shopping.