If you don't have your own Gold Chocobo, you'll need to defeat the Ruby Weapon before tackling the Emerald Weapon. You'll get the Desert Rose which you can trade at the Kalm Traveller for a Gold Chocobo. Then you can get the Knights of the Round needed to defeat Emerald Weapon.

After you've defeated Ultimate Weapon, Ruby will appear in the sand around Gold Saucer. Equip the Final Attack-Revive combination on all your characters, the Bahamut ZERO-Quad Magic on one, and Mime on the other two. Simply fly into him to initiate combat. DON'T ATTACK HIM UNTIL HIS CLAWS ARE BURRIED IN THE GROUND!!! If you do, he'll use his quicksand attack and you'll instantly lose a member of your party. Once his claws are buried, cast Bahamut ZERO on him and have the other two members mime it. If you do have Knights of the Round, you are better off not using because he will retaliate with Ultima, making short work of your party.
 

Once Ruby's down, exchange the Desert Rose for the Gold Chocobo and go get Knights of the Round from the Northeast Island. Also, exchange the Guidebook for the Underwater Materia (see the Kalm Traveller) You'll find the Emerald Weapon in the ocean as soon as you acquire the submarine, but if you attack him that early on, he'll make mince-meat out of you in about 30 seconds. When you're ready, equip the Underwater materia, Knights of the Round-Quadra magic, and mime on your two other characters. Do the same as for Ruby - cast Knights of the Round and have the other two mime it. When he dies you'll receive the Earth Harp as a reward.