In order to win the battles against most of the aeons easily, just go into the battle with all three of your characters as an Alchemist. You get the Alchemist dressphere by clearing out the fiends in the area that Clasko chooses for his chocobo ranch.

Once you're in the battle, just use "Mix" and mix two elemental Gems of the type that the particular aeon is weak against. Have each character to this and the aeon should go down before they can even really get off any good attack on you.

Ifrit - Ice Gem
Ixion - Water Gem
Shiva - Fire Gem

You can get these elemental gems by winning battles against various fiends that like to use attacks of those elements. Assuming you've been battling all around Spira, then you should have a fair amount of each of them in your stock.