The main rail stations are Termini in Rome, Santa Maria Novella in Florence, and Santa Lucia in Venice. Note that train sites use the names Roma Termini, Venezia Santa Lucia, Firenze Santa Maria Novella. I would suggest you look at Trainline which is very easy to navigate and offers both the Italian National Railway, Trenitalia, and Italo Treno, a private company. There you can see options for fares and times. The further in advance you buy, the more of a discount you can score. Discount fares do sell out, but trains seldom sell out completely. (Although I have friends who experienced a day when they could not get a train from Rome to Venice, so it does happen.) I recommend you buy AT LEAST a few days in advance.
If you are worried about how to get to the train station in Rome, take a taxi. With luggage, that's the easiest anyway, and Roman taxis are not expensive.
All of this is covered in the Rick Steves' Guide to Italy. You might want to pack it along for reference as it has a wealth of info especially for the first time traveler to Italy.