First: There is no such thing as Eurorail. There is only a train pass called Eurail, offering global prices for tourists and the European train network.
If you want to use the European train network, you talk about "taking the train". In Italy you will be using the trains of the national Italian rail company ferrovia or trenitalia (they change their name every few years as fancy strikes them).
The website is http://trenitalia.com/
I recommend that you buy point-to-point tickets, either online in advance or directly at the train station in Italy. Do not buy a Eurail pass, for your trip it will be cheaper to buy point-to-point.
There is no real alternative to taking the train, as flying is impractical and renting a car would be expensive and very stressful. Italians are the worst drivers in Europe and and you'd be completely mad to drive in Rome, Florence and Venice. (Rome is legendary for its mad inner city traffic, cars are banned in Florence's Old City and there simply aren't any roads in Venice.)
For hotels check out www.crosspollinate.com, it is a network of small private pensions.
Rolling luggage will be fine, but backpacks will work too. Take whatever you luggage you got, but make sure you pack light.
Days in each city:
Rome 3-5 days
Florence 3-5 days (if you are into Renaissance art spent more time in Florence than in Rome)
Venice 1-3 days
Naples 3-5 days (depends what you want to see, if it is only Pompeii you could stay for 2 days, but if you want Capri and the Amalfi coast too you need more time than 2 days )
So Happy...I am planning a trip for our honeymoon and I always wanted to go to Italy but there is so much info its hard to choose