Jim,
A Regionale Veloce train is an "express" Regional train that doesn't make as many stops. There are no price discounts for buying tickets in advance, so I'd suggest buying a ticket at Roma Termini using a Kiosk or at the ticket office, preferably the day prior to travel. One important point to note is that you MUST validate your ticket prior to boarding the train on the day of travel, or risk hefty fines which will be collected on the spot - DON'T forget! As the previous reply noted, there are usually no reserved seats on Regionale trains (at least not that I've ever seen).
For the trip from Vernazza (Levanto) to Varenna, a seat reservation on the Intercity trains IS compulsory. You may be able to save money on the Intercity portion of the trip with advance purchase, but at this late date the discounted tickets may already be sold out. You'll have to check and see if there are any Super Economy or Economy tickets left for the date and time you'll be travelling. You MUST be willing to commit to a specific train, date and departure time as the tickets are specific for one train only, and the discounted tickets are often non-refundable and non-changeable after purchase. To buy tickets for the Intercity, you might try www.captaintrain.com which is a very user friendly website.
When you arrive at Milano Centrale, you'll have to transfer to the Regionale for the final segment to Varenna. You should be able to buy tickets for that at the station in Vernazza but again, DON'T forget to validate prior to boarding the train on the day of travel. When you look at the electronic schedule board at Milano Centrale, you may not see "Varenna" listed as the destination as the end point of that train may be Tirano or elsewhere. Check the number of the train listed on your ticket.
