We're travelling from Amsterdam to Heidelberg next April. Will the price of our train tickets really be much higher if we purchase them at the train station on the day of in Amsterdam rather then months in advance? I remember when we were in Italy, the prices were actually lower the day of our departure. Any insights?
You can go to www.bahn.de and look up prices. You will see three priaces, two discounted ones, and one "flexpreis". The flexpreis is what you pay when you want a fully flexible ticket. It is also the price you pay at the station on the day of travel itself.
You do not need to buy tickets months in advance, but a few weeks in advance to get a discount is a usually a good idea.
Regarding Italy: I am not aware of any tickets getting cheaper last minute. Maybe you overlooked something.
You can go to the bahn website for you travel date and see the price of a nonrefundable ticket bought today (Super Sparpreis) and. Flex ticket, which is what you would pay at the station on your travel date.