This depends on what you assume of $\mu$, $\sigma$, and $f$. I'll provide a few examples. Let's assume $f$ is bounded throughout, since what I'll say below is still true in the unbounded case under suitable growth assumptions.

First, as long the SDE is well-posed (in the sense of weak existence and uniqueness in law), then the solution is a Feller process (this is somewhere in Stroock and Varadhan's book), and your function $g$ is continuous in $x$ whenever $f$ is continuous. If $f \in C^2$, then in fact $g$ is in $C^{1,2}$ (see Theorem 24.1 of Kallenberg's "Foundations of modern probability").

If you're interested in higher order derivatives, consult the vast literature on stochastic flows of diffeomorphisms. For example, if $\mu = \mu(x)$ and $\sigma = \sigma(x)$ are time-homogeneous and $C^\infty$ with bounded first derivative, then $g(t,\cdot)$inherits derivatives from $f$; i.e. whenever $f \in C^k$ then $g(t,\cdot) \in C^k$ for each $t$. (See Theorem V.13.8 of Rogers & Williams, "Diffusions, Markov processes, and martingales" vol 2.)

Finally, if you are really only willing to assume $f$ is measurable, the story is a bit more complicated. You'll want uniformly nondegenerate volatility, i.e. $\sigma\sigma^\top \ge \delta I$ for some $\delta > 0$, so that the noise helps you recover some smoothness. In this case, at least if $\mu = \mu(x)$ and $\sigma = \sigma(x)$ are time-homogeneous, then the map from the initial condition $x$ to $\text{Law}(X^x_t)$ (the distribution of the SDE solution at some time $t > 0$) is typically continuous in total variation. See sections 11.3 and 11.4 of Stroock and Varadhan's book.