You might consider your problem in a generalized setting. Let $H$ be a Lie group and $X$ an $H$-manifold on which $H$ acts freely and properly. There is a unique manifold structure on $X/H$ for which $X\rightarrow X/H$ is a principal $H$-bundle (and in particular, a fiber bundle). If $H$ is a closed subgroup of a Lie group $G$, then $H$ acts freely and properly on $G$ by left-multiplication. You then obtain the desired result.

I have never before attempted to construct explicit local cross-sections. You might try to use the exponential map $\exp:\frak{g}\rightarrow $G$ $ (a local diffeomorphism) together with your surjection $\frak{g}\rightarrow\frak{g}/\frak{h}$ to construct a local section at the identity. Otherwise, I think you might need to consider the Implicit Function Theorem (not particularly explicit).