Italy is wonderful, gorgeous and one of my favourite places.
Generally for flying in comfort, take a change of underwear and clothing in your hand luggage, also wet wipes, plenty of books and some sweets. Check the airline website for information on restrictions on carriage of liquids and prohibited articles. Make sure you know when you need to be at the airport and allow extra time to get through security.
Tuscany is incredibly wonderful with great food and lovely people. San Gimignano has some beautiful towers, picturesque cobbled streets and really good food and wine. If you're hiring a car then you can drive into the chianti countryside and enjoy some wine tasting. Look for the small establishments with "vendita direta" outside as these are places that sell wine to the public. Buy some and take it back to the villa.
Siena and Firenze (Florence) are both lovely. Personally i think Siena is more intimate and unspoilt. Firenze has an awesome leather market in the Cascine park which is great during the day but a bit dodgy at night. Firenze also has the Uffizi gallery which is worth spending time in. Most tourists also like the Leaning Tower of Pisa but I wouldn't bother, it's not much to look at and full of tacky souvenir stands.
Venice is beautiful but very expensive. Any drinks on Piazza San Marco will cost you over the odds. Don't hail a gondola and expect it to be affordable. If you stop one they will charge you over the odds and subject you to bad singing. The cheapest way to get the experience is the traghetto which is essentially a ferry service at some points from one side of the river to the other. It's not as romantic but it's a lot cheaper. Water buses are on the other hand very affordable and a lot of fun. If you take a water bus out to Murano on one of the small islands you get to see the views and then see the glass museum.
Rome is lovely but watch your wallets at all times. There are a lot of pickpockets. Don't wear any ostentatious jewelry and keep a sharp eye out especially on the buses. My favourite art gallery there is the Doria Pamphili which is in a beautiful building and has some lovely unusual collections. People like throwing coins in the Trevi fountain and seeing the Spanish steps.
I didn't go to the Vatican but I understand it's quite large and can take ages to get around. If you want to see the pope do the thing where he stands on the balcony (not sure the correct name) and blesses people, check in advance for the timings.
Other things that are fun are window shopping and eating nice food. In Italy I'd say avoid anything that has plastic menus in several languages displayed and waiters walking outside trying to drag people in. Locals don't eat there. In the early evening the main street of any town will have what they call the passegiata which is where people dress up and walk down the street, meeting their friends and being seen. A good way to find somewhere to eat is go to a pavement cafe, watch the Italians passing and see where they wind up going to eat. Italians do tend to dress up for dinner and operas and tend to eat dinner quite late. Gelato (ice cream) is heavenly, pizza is great and pasta is so good you'll dream of it later.
Opera is wonderful in Italy. Venice has a lovely opera house, so do most other places. Nothing beats the sight of Italian opera in the native language.