Don't worry, the birds really caught me off guard as well. Either stick to forest cause you usually only fight Goblins and the Porcupine monsters or find the Chocobo forest in the southwest forest and then ride it to the Mist Cave. You don't have to fight the Eagles in there so you can get some real levels and gil for stuff. Once you get a few levels on your belt its' easier to deal with the birds cause your party isn't so slow that gradual petrify only allows you one turn.

Just stick to it and stay optimistic, eventually only the bosses become issues until you get to the final dungeon.