I was exploring Gran Pulse and we found this little green cactus thing that is stuck in a running position. there was a little cutscene with Sazh, I found one and fought him and won, so I thought he was some sort of story related thing... but the battle just ended. Then I found some more around the Central Expanse. What is their purpose or do they have any?

Play more Final Fantasy. Seriously. 
 
 
 
:P

Sabotenders (Cactuar I think in English) are a mainstay of the Final Fantasy franchise, if you haven't played many of them it may seem odd to you.  They are one of the typical FF monsters though, along with things like Tonberries, behemoths, et all. 
 
They drop chocobo plumes or cactuar dolls if I recall correctly.

might i point you in this direction: cactuar

Haha, thanks. The FFs I've played and beaten were 3 (DS), 7, and 10. I don't remember a Cactuar in any of them... But I do remember tonberries from 7, the guys who slowly get closer to you. I wonder if they're in this game... :P I suppose the battle system of this game would mess it up though.

Yes, Tonberries are in this game. 
 
Sabotender were in FFVII but extremely rare, and I believe they were in FFX (I know they were in FFX-2)