Airbuster (fought outside No. 5 Reactor after a brief meeting with President Shinra) - Enter the battle with Cloud and Barret's limit gauges full. When the Airbuster is facing away from one of them, have them use their limit break. From behind, Braver and Big Shot cause over 600 points of damage each to the Airbuster. The combined power of the two can destroy the Airbuster, which has about 1200 HP, in two hits, provided he is facing away from whoever uses the limit break.
Gi Nattack (fought in Cave of the Gi) - This one's easy. If you have an X-Potion, use it on the Gi Nattack and it will kill him. If you don't have an X-Potion, five or so Cure 2 spells should be enough to take away his 5500 HP.
After Meteor has been summoned, return to Cosmo Canyon and visit the Item Store (to reach it, climb up the ladder to the far right outside on ground level). There was previously a rope barring a path that lead around to the back of the store, but now that it's disappeared, follow the path around. You'll find yourself in a room with the "Fullcure" materia and two other items.
If you don't find Cure 3 or White Wind effective enough, you might like to use Fullcure.