There's supposed to be no evil in Valinor (at least, not since Melkor left it).
Given that, there's presumably no need for elves to learn to fight. They can spend all their time on the poetry, music, and dance that they so enjoy. But if this were the case, then the Battle under the Stars would never have happened, since Melkor would've wiped the defenseless Noldor out.
How did the elves learn to fight in Valinor? Only thing I can think of is that they learned from the Valar (e.g. Tulkas, who was after all fond of 'contests of strength') - but such contests are presumably still not lethal arts.