I'm going to be in Italy late September through early October and I was wondering if there was anything I should book in advance besides lodging and airfare?

We will be going to:
Rome
Almafi Coast/Pompeii/Naples
Tuscany/Florence
Venice

I heard trains would be good (what's the best website?), anything else? We plan on going to the Vatican and doing just standRd touristy stuff such as sight seeing, tours around areas (we prefer solo vs groups) and museums.

To avoid very long lines get advance tickets for:

Rome
Vatican Museum
Borghese - if you want to see this gallery
Colosseum

Florence
Uffizi
David at the Academia

Venice
Doge's Palace
Duomo

You can avoid long lines for the Colosseum by buying tickets at the entrance to the Palatine Hill--Palatine, Roman Forum and Colosseum are all on the same ticket.https://www.rometoolkit.com/whattodo/roman_forum.html and not so many people try to get in there.

Otherwise concur with ny traveler

Trains are very good in Italy. The best site--which offers booking tips, etc., is:

http://www.seat61.com/Italy-trains.htm#.VcIeY7eD7cs

You may want to make dinner reservations, especially at popular restaurants. There's nothing more disappointing than to arrive at a place only to find out that is full.

In Florence, I'd add the Brancacci Chapel to the list. If you wanted to have a guided tour of Palazzo Davanzati, you need a reservation. The lower floors (including the lace museum) are accessible without reservation.

In Rome, the Scavi Tour (excavations under St. Peter's).