Well I guess it really depends on how much you expect and where you want to swim.
For a little splashing around the sea I think you might learn it pretty fast but then you shouldn't swim too far away from the beach. Most people I know learned swimming as a child :S so I can't really tell if the learning curve is a lot different being an adult. Probably some weeks for the basics and some more if you are really dedicated and want to improve.
What I'd suggest you to get is some fins (not tooo short ones). It's much easier to stay on the surface with those and the only thing you need to mind is that you don't bend your legs while swimming (you won't drown if you do... it's just much easier if you keep your legs straight!)
And if you get a snorkel aswell you can even breathe a little with your head underwater if there's only some smaller waves.
Thanks for the tip @vandaze! I'll probably look funny with those fins but I gotta do what I gotta do lol. I definitely have to learn tho!
Yeah I mean... if you're invited for a pool party or sth. like that it's probably not really helping with the fins :D but travelling to the caribbean and swimming in the sea there are probably a lot of people doing some snorkeling with fins.
BUT of course you should first at least learn to swim well enough to get back to the beach if you should ever loose one of your fins ;)
I bet there are some courses in your town you might join in to learn swimming?!
Yeah there are! But I'll probably be the oldest lad in the class with little kids lol. I'll probably ask one of my friends to teach me, some have a back-yard pool too! It would be better to learn from a professional, but maybe I can watch some youtube videos on the side :)