Train from Pisa to Vernazza

No need to purchase in advance. For some categories of train (intercity or frecciabianca) buying in advance would give you a chance to get a discount. However discounted tickets are non refundable if you miss that train, therefore you would lose some flexibility. Since the distance is not that long and the discounts are therefore marginal, I would keep the flexibility and pay the full price on the spot when I show up at the Pisa station.

roger,

I wouldn't bother purchasing advance tickets for that route. It's possible that one leg will be via InterCity or FrecciaBianca, but that's a relatively short distance so the cost of "Base Fare" shouldn't be too high. It's more likely that you'll also be using a Regionale train for one leg (DON'T forget to validate!) and it's not easy to purchase tickets for those in advance nor is there any price break. Just buy the tickets at the station using one of the Kiosks or at the staffed ticket office.

We found long lines and broken ticket machines at Pisa when we were heading to Manarola. It took more than 30 minutes to get the tix and we only made our planned train because it was late. If you are in Pisa the previous day, you could go to the station and buy your tickets then so you don't have to stress on your travel day.

Thanks all, for the helpful comments. I will take your advice!

I think it depends if you need the flexibility or not. We also ran into lines and machines that rejected our card a few times before it would read it, so just allow a bit of extra time as a precaution.