I'm looking for the fastest and the cheapest way to get from the center of Amsterdam to the center of Rotterdam for three persons. Should I take the train? A bus? Even a plane?
The fastest way (41 min from station to station, half-hourly on weekdays) is the IC Direct (high-speed train), but you need to buy an additional ticket on top of your regular ticket (€16.80 = 14.50 + 2.30). The normal train connection (60–70 min, 2 trains per half-hour) would cost you less (€14.50). You can look up times and prices and buy tickets on the NS website. You will be charged €0.50 extra if you pay with credit card.
If you are able to find a regular Dutch traveler, you can save 40% on your ticket. Holders of the Dutch railways reduction card get this reduction outside peak hours and can take up to three fellow travelers with the same reduction. So to go cheap you need to make Dutch friends ;)
A plane is certainly not the answer, since the two airports (AMS and RTM) are only 45 km apart. I checked Kayak anyway; there are no nonstop flights. Air France will fly you AMS-RTM via London (LCY) for EUR 242 and it takes 3 hours including the layover. So that is neither faster nor cheaper than a train, not by a long shot!