No one collects or checks baggage as you board, so there are no fees. You carry it on board yourself. There are luggage racks above seats, but big bags may not fit there. There are luggage racks at the end of train cars where people tend to leave bigger bags. Unless you have some strange super-oversized bag that no one really travels with, don't worry about the size.
Some people may worry about having their bags stolen. Shouldn't be an issue on these express trains with only a few stops - but you could try tying it up to the racks if you are worried about it.
I always go second class in Italy - on that train, it's still a nice seat. Some people prefer to pay a little more for first class; I only did that in Poland where the trains weren't very nice, and 2nd class was kind of like "3rd class," and real first class seats weren't expensive. 2nd class may be a little more crowded but probably not an issue on trains with reserved seats. Really up to you.