Buying train tickets on-line vs at the station
 

Use the trenitalia website and search for your journey (using Italian place names). If it's a regionale service no need to pre-book. If it's not - and you know when you'll be travelling - then likely the prices will be cheaper if you book in advance, but they have strict change and cancellation conditions.

Milano to Varenna is a regional train. Price is the same no matter what. What you gain with an online ticket is you don't need to line up to buy it or to validate it. But price is the same.

The second one is a connection. The first part is the same regional back to Milano and no saving buying it in advance. The second you can save money buying in advance.

Not sure why you're worried about missing your train

If you buy a regional ticket online, it's good for the train you requested or another regional train on the same line departing in the next three hours. The ticket is not refundable or exchangeable.

If you buy the ticket at the train station, it's good for the date you requested. You have to validate it in a machine by the track, and at that point it's only good for the next three hours. If you neglect to validate the ticket, it's as if you have no valid ticket at all, and you're subject to a heavy fine. Before it's been validated, you can exchange it or get a refund if you decide not to use it.

I prefer to buy regional tickets at the train station, for the greater flexibility. You can buy them ahead of time, when you happen to be at a station, but be sure to specify the date you want to travel. And be sure to validate them just before boarding the train.

Milano - Varenna - Milano: as above

Milano - Rapallo (OW):
direct Intercity train:
Standard fare (available at any time): 24 EUR
Super Economy Fare if you book up to 3 months ahead (valid only for the train you booked): 10 EUR
Regionale trains (only slightly slower, but with train change at Genoa):
standard fare 16 EUR (available at any time).

Thanks everyone for your responses and suggestions
 

Thanks everyone for your responses and suggestions

In Varenna there is no 'manned' ticket booth, just machines. There is a cafe at the train station that does sell train tickets, but of course only when it is open. Ticket machines can sometimes be broken so if I know I'm departing from an unmanned station I usually get the ticket ahead of time. But you could buy your ticket both to and from Varenna at Milano when you get there. The return ticket is good for any day until you validate it so you can get it several days in advance and just hold on to it and then not worry about having to buy it in Varenna.

Isabella, as of this year, regional tickets are issued only for a specific day, so you have to specify what day you want to return. Then it's good for any time of that day, until you validate it. Once validated, the ticket validity expires in three hours.

Do not forget to validate! You're considered to be travelling without a ticket if it's not been validated.

As mentioned above, tickets bought online don't have to be (and can't be) validated. They're already sold as valid for a specified three-hour period.