This question originates from me and a friend watching 300: Rise of an empire (I know, I'm sorry). He was adamant that none of the Greeks falling in the water could've survived because surely they wouldn't have gotten swimminglessons. This seemed unlikely to me, looking at current navys.
So all around; What was the training of Greek soldiers in that era like?
Bonus question; How likely is it that Greek sailors were tought how to swim?
I'm afraid your friend is mistaken. Herodotus specifically says that at Salamis the Greeks suffered few casualties from men thrown overboard, noting that they were able to swim and that those who were not killed outright simply swam to Salamis. Herodotus then notes that the Persians lost quite a lot of troops (presumably mostly their marines--the rowers probably would've been Phoenicians) thrown overboard, specifically because they could not swim. How your friend might think that the Greeks, could not swim is beyond me. Almost every Greek city was near the coast and many were islands. Swimming is something that Greek children would've learned at an early age, and it's mentioned as early as Homer
Greece was well known for having swimming pools and living on the ocean there is no reason they would have avoided learning to swim.
http://cev.org.br/biblioteca/swimming-and-rowing-in-ancient-greece/
Drowning is even pretty rarely mentioned in greek literature:
(Discussion: This historical review reveals that although swimming was considered a common activity in ancient Greece, drowning was an unavoidable fact of life as reported in both mythological and historical narratives that were reviewed by Devouros, sometimes in the form of oral history, anaglyphs, or angiographies.)