Let $G$ be a Lie group, and $X$ a homogeneous space of positive dimension, say $X=G/H$ for a proper subgroup $H$. I'm interested in understanding the group cohomology $H^1(G, C^\infty(X))$, particularly in the case $X=G$, defined as the smooth 1-cocycles $\phi:G\to C^\infty(X)$, modulo coboundaries, with $C^\infty(X)$ considered as a $G$-module.
I currently understand this group mainly in terms of cocycles modulo coboundaries, but I'm missing a good intuitive picture, as well as concrete computational tools.
For example, is it obviously the case that $H^1(G, C^\infty(X))=0$ when $G=X=(\mathbb{R}^n,+)$? For other $X$, it is obviously the case that $H^1(G, C^\infty(X))\neq 0$? This seems to be true from the basic cases I've looked at, in particular $n=1$, but I'm not sure how to approach it in general, or how to fully characterize $H^1$ in these cases.
And are there standard methods for constructing homogeneous spaces with an interesting $H^1$, when $G$ is one of the classical Lie groups? What happens when $G$ is compact and $X=G/T$, where $T$ is a maximal torus?
I do have Knapp's book on cohomology of Lie groups in the mail, but if there are good online references for this kind of thing, they'd be much appreciated.