Let $S = \oplus_{n \geq 0} S_n$ be a graded ring. Let $S$ be a finitely generated $A$-algebra, where $A = S_0$, a commutative ring with unity. Then there exists $t_1, .., t_M$ homogeneous elements of positive degree that generate $S$ over $A$.
It then follows that $S$ is isomorphic to $k[x_1, ..., x_M]/I$ as rings for some ideal $I$. Does it follow that $I$ must be a homogeneous ideal of $k[x_1, ..., x_M]$? I am guessing it is the case, but I just wasn't sure. Thank you!
Only if you correctly define "homogeneous". For example, take $R = k[X^2, X^3]$. Then $R$ is isomorphic to the quotient $k[s,t]/(s^3 - t^2)$, and $(s^3-t^2)$ is not a homogeneous ideal according to the usual definition.
However, it is still homogenous if we put a different grading on $k[s,t]$, namely we specify that $s$ has degree $2$ and $t$ has degree $3$. We should do this anyways, if we want $R$ to be a quotient as a graded ring, rather than just as a ring.
Given this definition, here is how to see that $I$ is homogeneous in general: Suppose that $f \in I$, and $f$ equals the graded sum $\sum_i f_i$. Then $\sum_i \overline{f_i} = \overline{f} = \overline{0}$. Since each $\overline{f_i}$ has different degree, we have $\overline{f_i} = \overline{0}$ for each $i$, so $f_i \in I$ for each $i$.