From the airport I think your only options are flat-fare taxi, probably 2,500 + THB, or private car/limo (more).

From Bangkok there are infrequent trains (Hua Lamphong), frequent buses from the Southern Bus terminal, and the mini-van option from Victory Monument. So you could take the 551 bus (or a taxi) from Suvarnabhumi to Victory Monument (34 THB, typically quick although sometimes you're on the expressway, sometimes on the surface road {Rama 9}), and then the mini-van to Cha Am/Hua Hin (~ 150 THB). Returning is more challenging as the mini-van might full ex-Hua Hin by the time it gets to Cha Am, but think you can make a reservation in advance. Not sure about local transport in Cha Am, to/from the place where the mini-van drops you off. I assume they have song taews?

Or you could take the 556 bus (or a taxi) from the airport to the new Southern Bus terminal, and then a bus to Cha Am (Hua Hin) from there. Buses depart to Cha Am every 30 min.