Let $K$ be a simplicial complex and $K^r$ its $r$-skeleton, I wanto to show that the simplicial homology groups $H_p(K)$ and $H_p(K^r)$ are isomorphic for all $0 \leq p \leq r-1$.
And also I wonder if there is a relation between $H_r(K)$ and $H_r(K^r)$.
To answer your second question, first note that $H_r(K) = H_r(K^{r + 1})$. Now consider the pair $(K^{r + 1}, K^r)$. We get an exact sequence
$0 \to H_{r + 1}(K^{r + 1}) \to H_{r + 1}(K^{r + 1}, K^r) \to H_r(K^r) \to H_r(K) \to 0$
This is telling you how to get $H_r(K)$ from $H_r(K^r)$ -- you have to quotient by the image of $H_{r + 1}(K^{r + 1}, K^r)$, which is a free group generated by all the $(r + 1)-$simplices, so that you're modding out by simplices one dimension larger that fill in the holes.