We say an $R$-module $M$ has finite flat dimension (finite projective dimension) and write $fd_RM<\infty$ (resp, $pd_RM<\infty$ ), provided there is a finite flat (resp, projective)resolution
$0\rightarrow F_n \rightarrow F_{n-1}\rightarrow \cdots \rightarrow F_2 \rightarrow F_1 \rightarrow F_0 \rightarrow M \rightarrow 0$.
If $R$ is local, it is well known that every finitely generated flat module is projective. Therefore in case that $R$ is a Noetherian local ring for every finitely generated $R$-module $M$, the concepts of flat dimension and projective dimension are equal.
Now, let $(R,m)$ be a commutative Noetherian local ring and $M$ a finitely generated $R$-module with $fd_RM<\infty$. It is clear that $M$ is a finitely generated $R/Ann_RM$-module. I want to show that $fd_{(R/Ann_RM)}M<\infty$ but I dont have any idea for it.
This is not true. There are several ways to construct an example. For instance, take $S=k[\![x,y]\!]$ and let $R=S/xy$. Consider the $R$-module $M=R/x \oplus R/y$. It is easily seen that $M$ is faithful over $R$, and so $\operatorname{Ann}_S(M)=(xy)$. As $S$ is regular, we have $\operatorname{pd}_S(M)<\infty$. But neither $R/x$ nor $R/y$ have finite projective dimension over $R$. Indeed, the complex $$\cdots \rightarrow R \xrightarrow{\cdot x} R \xrightarrow{\cdot y} R \xrightarrow{\cdot x} R \to R/x \to 0$$ is a minimal free resolution of $R/x$ over $R$ and shows $\operatorname{syz}^1_R(R/x) \cong R/y$.