1) Set your expectations, unless you are in first-class on a high-speed train, do not expect your ride to be cushy. This way you will not be let down by the conditions and you may sometimes find yourself pleasantly surprised. First-class is sometimes worth the extra money because it is always less crowded, air-conditioned and has an outlet or charging station, and sometimes even free wifi service!
2) IC (inner city) or R (Regional) lines are what the everyday person uses to get around. They may or may not have air-conditioning so always bring a hand fan with you, especially if you are traveling in the summer time. The Circumvesuvius line is definitely never air-conditioned, and it can get crowded and stuffy, so a fan may save your life.
3) IC and R trains are also almost always running late, so do not depend on them getting you anywhere on schedule. And if you have a connection, always have a backup plan in case you miss it due to the train coming in late. If you do miss a connection and it is the fault of the train, simply talk to someone at the ticket counter and they should be able to get you on the next train out at no additional charge. I would actually recommend not having a connection at all and just buying your tickets on the spot when you get to each station.
4) Do not listen to people when they tell you to book ahead because seats get sold out. I have been traveling at peak summer season and I have never not been able to get a ticket on the spot, literally minutes before the train is set to leave. If it makes sense for you to book ahead by all means go right ahead, it won't hurt, especially if you are traveling in a group and want to sit together, but don't feel like you need to do it.
5) If your ticket requires validation, use the green or yellow machines at the station to validate it BEFORE you get on the train. This is for any ticket that is valid for a set period of time and is not for a specific train, with a specific seat. R train tickets usually need to be validated. If you are caught without a validated ticket you can be heavily fined, or kicked off the train and your ticket taken from you so you will be forced to buy a new ticket to get where you need to go. (This happened to me and I was not a happy camper)
6) For some trains the prices change inexplicably. A train from Florence to Pisa may have cost you 3.10 in the morning, but the ride back in the evening will cost double. Don't bother questioning it, it is a mystery of life that may never be solved.
7) Know ahead of time when you are set to get to your destination station, and know the stops leading up to it. Very few trains actually announce their station stops, and some of the stations are very poorly signed, so you may end up missing your stop if you are not careful.
8) Good luck, and god bless!