Today I took my puppy -15 month- to swim for the first time. He got the idea really fast but kept on swallowing air. He ended vomiting and burping all of it. Is that something that he will figure out as he becomes a better swimmer?

Yes. Just make sure that if he swallows too much air/water while swimming that he is not too far out so that if he gets in trouble you will be able to help him. Not likely that this will happen, though.
Let him free swim without throwing anything for him so he will relax and learn how to swim properly before getting him jazzed up.

Another recommendation is that you don't feed them right before they swim. If he is just starting to learn how to swim, and he is swallowing alot of water and air - then that is not good on his system.