They load the whole train on a barge and take it across the water. It was awesome.
Ride 1st class, it is worth every penny.
We rode the train from Naples to Messina. We started at night, and it was daylight for the southern part of our trip. It was absolutly beautiful.
I am not sure, but think that you will have to transfer at Messina for the balance of your trip.
Not sure about buying the tickets, but i was told that one of the best places to buy them is at a travel agency in Italy/Sicily prior to your train trip. I would buy them as soon as I arrived in Sicily.
The TI's (information booth) at major airports/ train stations are really helpful, and provide great info about the train system.
Don't forget, ther are lots of lo-cost airlines that ply the Catania to Naples route.
If you need more info about this, post the same question on the Sicily board on Trip Advisor.com.
Have fun...it is a great trip!