Spend a bit of time on the trenitalia.com website to look at your options. Just click on English in the upper corner. Put in the cities and be sure that the time shown is early in the day. When the routes appear, select a time, click on the circle, scroll down and click on details. You will learn about the types of trains, whether they need reservations, etc.
Manarola does not have a manned station so you need to buy all your tickets in Lucca.
Many of the routes between Lucca and Manarola have only regional trains, which do not have first class. Some of the time there is an IC (Intercity) train for an hour that would have first class. But there's really not much difference between the 2 classes.
I have not been to Montalcino, but my guide book says there are no trains, only buses from Siena. You can take the train from Manarola to Siena and again some routes are regional only and some have one segment with a train with first class. There are 3 train changes just to get to Siena, then a local bus ride up to the bus station. I'm thinking you should consider spending a night in Siena.
Regional trains do not have reserved seats and you can use a ticket on any train on that route for the next several days. Other trains do have reserved seats and the ticket is only good for that train. When we traveled we never saw a full train - you don't need to worry about them selling out unless it is a special celebration or something like that.