Well, at first he wanted the sword...then...uh...I dunno.
This is my first time watching Inuyasha.
It has hinted that Naraku isn't a full demon, so it makes sense why he's searching for the jewel, to become a pure demon.
Inuyasha says he wants the jewel for the same reason, but it has also hinted that the jewel could turn him into a pure human too (I'm guessing this is the ultimate outcome?)
I know Sesh wanted Tetsusaiga, but now he's going to get another sword, but I'm not sure what he's ultimately trying to do. I guess the jewel could make him stronger so he's looking for it too. He has tried to Kill Inuyasha before, I don't know if that's the ultimate plan or not.