Inuyasha still feels indebted to Kikyo since he promised her his life and that he would go to hell with her. Kagome often gets either very sad or jealous when he goes to see Kikyo. And Inuyasha, in return, often says something very stupid that worsens the situation.
Another struggle or problem that Inuyasha has to face is: what he wants to do with the jewel as well as finally choosing between Kagome and Kikyo.
Although Inuyasha never directly states it, or despite his best effort to hide it, he does care a lot about Kagome and he doesn't like the fact that there's 500 years between them either.
Kagome's struggles are also hard and angsty. She has to decide whether or not to stay in Sengoku Jidai when the quest is over, assuming she lives through it.
On one hand, should Inuyasha choose not to go to hell with Kikyo and stays in the village; then she would stay back in the past with him, also assuming he chose her over Kikyo. On the other hand, he could go to hell and Kagome would have to decide if she wanted to stay in the past where it reminded her of Inuyasha.
Kagome also has to deal with the fact that she just wants Inuyasha to be happy. She just wants to stay by his side until the end, when he's truly happy with what he has. So, being the unselfish person she is, she wouldn't stoop so low as to jeopardize any chances Inuyasha might have at happiness, and that includes Kikyo and the jewel.
Another thing to think on would be the whole going-to-hell-with-Kikyo thing. Inuyasha wouldn't be going to hell with Kikyo's soul---he'd be going to hell with Kagome's soul.
If you all remeber correctly, Kikyo is living off a small portion of Kagome's soul---not her own. Kikyo's also feeding off the souls of the dead, as her snake-like-soul-catchers bring her the souls of young maidens. So, I guess Inuyasha has got a lot of contemplating to do, doesn't he?
