There are two train companies operating in Italy: Trenitalia (the national company) and Italo (smaller, newer, private company). Italo provides trains on some of the same routes as Trenitalia.
The Trenitalia site is for Trenitalia trains only. Italo has its own site - I've never used it or them but would if they had better prices on routes I wanted to take.
The Trainline site is a third party site that should be the same price for Trentailia trains as the company site. It also includes the Italo trains. So you could use this site for everything - prices should be the same. Some people prefer using the company sites directly instead of a third party booker like Trainline. But it probably doesn't matter much.
I don't know the best routes. I'd use the websites listed above and check various connections. You might also plot cities on a Google Map (even though you won't drive) to get an idea of the most practical links distance-wise between cities - I do that too when I plan trips.