I disagree a bit with Jmat, but this may be a difference between American and Australian English.
To me, "going to THE sea" is emphasizing going to a particular body of water: "Moses went into the Red Sea."
"Fishermen going to sea" sounds correct, and emphasizes that they are going on a voyage or, in this case, out on the water to go to work.
The second is correct, in my opinion.
The last, as before, should be "They just go to sea to fish" seems correct to me.