college graduation trip to Italy itenary

Nice graduation present, Pam!

You are correct. By train is the quickest and sanest way to travel. 2 hours Venice to Florence SMN station, 90 minutes Florence to Roma Termini station. A breeze.

Lodging: for your kind of money ($150-200/night), that time of year (high season), you might want to investigate VRBO or AirBnB for an apartment, if you don't want to go the pensione route. A decent hotel in any of those places in a desired area will set you back over $200 for sure. But you have plenty of time to survey the market. In Florence, you should be able to stay within walking distance of the train station (easy for day trips), but not sure about Venezia Santa Lucia or Roma Termini stations--not that it's mandatory that you do so, but it certainly would be easier in Venice for your Verona daytrip.

Can't vouch for Venice re: tours--someone else will have to. Wineries in Tuscany, a private tour (or part of one) that originates in Florence would be your best bet, as trains/buses out of there into the Tuscan hills aren't that prevalent. On your own, you could easily take a day trip to Siena by bus and do the wine & food tour there--I have heard great things about it. When we were last in Rome, we utilized the Hop-On-Hop-Off bus to tour the attractions. It's a no-brainer at around 20 Euro per day (in 2010).

How to split the time, assuming it's 12 days, 11 nights?? Even with the Verona trip, I'd do 3 nights Venice, 4 nights Florence, 4 nights Rome. Personally, I'd spend more time in Florence but you can't give Rome short shrift either. A good dilemma to have, no?

Enjoy your planning!

Yes, trains are the way to go on this trip. A day tip to Verona is very doable. Get an early start and return late and you should be able to cover Verona fairly well, but not thoroughly. For Hotels we like Pensione Gueratto in Venice; in Rome we have stayed at Hotel Fontanella Borghese many times and have always been pleased. While it is not "luxury", it is very nice and the location is great. The staff is very friendly and helpful. You can walk to Spanish Steps, Pantheon, Nuvona, even Vatican city. They also have a sister hotel closer to the Pantheon called Hotel Due Torri. We always send an email directly with our request for a booking and we get an answer back in 24 hours or less. Ask for a room overlooking the courtyard as rooms overlooking the side street can get noisy. The web site is www.fontanellaborghese.com. For tours, we have always used the self guided tours in the RS guide books and have not been disappointed. With only 12 days including travel time, I would just split the time between the three cities but would not try to do more or it will all just become one big blur. RS has a tour of those three cities, but since you want to do it independently check out his tour itenerary for that trip and alter it to fit your own desires.

If it is just you and your daughter, I recommend Orsa Maggiore hostel for women in Rome. Location is perfect - in Trastevere with an easy walk to Pantheon in one direction and Vatican in the other. Not luxurious, but very reasonable cost. I am headed there tomorrow for my third stay.
In Venice I like either Do Pozzi or Hotel Flora. Both are usually under $200 and, since Venice can be challenging to traverse on your first trip, it is good to have a hotel near a major landmark. They are both off the major street heading away from San Marco.
In Florence I stay at the Duomo Hotel. It is located literally adjacent to the Duomo on the 5th floor of the building. Staff is very cordial and rooms are quite adequate.

Each of these cities has more than enough to keep you occupied for months, so do your research and choose the sights you most want to see on this trip. Remember that you can make more than one trip to Italy so don't try to fit everything in. On the other hand, do give priority to your areas of interest.
There's been a lot written recently about "experiences" being more memorable than "things" - this trip will provide an experience for both you and your daughter to remember forever. Great choice of a graduation gift!

Effectively you've got 10 full days, which I would divide in 3 days for Venice, 3 for Florence and 4 for Rome.
Would you need a tour to make the most of your stay? Venice is a place to get lost, to find something you didn't expect, and to find your bearings again. Florence has an easy to navigate compact city centre, and you know best what you want to see (and not). The attractions in Rome are more spread out. What you need is a decent guide book, a good map, and to know how to use bus 40 and 64, which run from Termini station to the Vatican, closely passing the Forum and the Piazza Navona, which you sooner or later want to visit.
A winery tour could well fill up one full day of your time in Florence. That day could also be fruitfully spent on a daytrip to Pisa and Lucca (less than 1,5 hrs from Florence, and Pisa another half hour further on).
I would say that your budget is good enough for a decent hotel in a decent neighbourhood. (I once had a most comfortable, pleasant and friendly hotel near the Termini station in Rome for EUR 180, which is within your price range.)
For train tickets: it pays to buy them well in advance - two to three months - on http://www.trenitalia.com/tcom-en. It's easy and reliable with paying by credit card and printing at home. The site speaks English, but only understands the Italian originals of the stations you need: Venezia S. Lucia, Firenze S. M. Novella, and Roma Termini.

Do you have the Rick Steves Italy book yet? It will be the smartest money you spend on your trip. It will help you figure out what your itinerary should be, based on your specific interests.

If you do 3 nights in Venice, that's only 2 full days. If you spend one of those days in Verona, you have only one day in Venice. And you may have some jet lag to deal with those first few days.

I'd say either increase Venice to 4 nights or skip the daytrip to Verona.

12 days total = 11 nights total = 10 nights in Italy (with one night on the plane)

Day 0 Depart Home
Day 1 Arrive Venice (3 nights)
http://www.violinodoro.com/
Day 2 Venice - San Marco and Doge's Palace
Day 3 Venice - Grand Canal and other sights
Day 4 Train Venice to Verona (for sightseeing), then to Florence (4 nights)
http://www.hoteldeimacchiaioli.com/
Day 5 Florence - Academy for David, then to Siena
Day 6 Florence - Duomo and other sights
Day 7 Florence - Uffizi, then to Pisa/Lucca
Day 8 Train Florence to Rome (3 nights), afternoon in Borghese Gallery
http://www.hotelsantamariatrastevere.it/
Day 9 Rome - Colosseum, Forum, Trevi Fountain, Spanish Steps
Day 10 Rome - St Peters, Vatican Museums, Pantheon, Piazza Navona
Day 11 Depart Rome

For hotels, start thinking in euros instead of dollars. At current exchange rates (found easily through Google or http://www.xe.com), your budget is about €130-175 per night. That is doable, with some research. Start with Rick's listings, and then double check them on TripAdvisor. Be aware that for any place in his books (or popular for any other reason), you should book earlier rather than later.

Congratulations on your plan to fly into Venice and out of Rome! You've already made a wise choice that so many don't (they fly round trip to Rome, then lose time getting to Venice).