Do aeons lvl, or get xp?

[Charlie Friggn Brown]

I mean do they get any better from fighting more often?

[Moridin]

untill you gain the ability to upgade individual stats from Belgamine on the calm lands, the Aeon's stats are dependant on Yuna's stats.

[Quackberry]

They don't level or gain experience but they do get stronger the more battles you've been in. From your 60th battle, every 30 battles will increase your Aeon's stats up until the 600th battle. This is any battle you've been in and even counts one before you have Yuna in your party and even those you run away from.

Their stats also increase as Yuna's stat increase but it's not as basic as simply being +4 Strength for Yuna = +4 Strength to your Aeon's (it does work this way for Luck though, Yuna's Luck = Aeon's Luck). Instead, raising a stat on Yuna will contribute towards your Aeon's same stat but also to it's other stats to a lesser extent (except Luck).

The other way is to directly increase their stats using that item from Belgemine you can obtain in the later parts of the game.

Here's an FAQ that goes into a lot of detail about Aeon's stats.
http://www.gamefaqs.com/ps2/197344-final-fantasy-x/faqs/21367