One can approach the study of diffeomorphism groups from many perspectives: topology, geometry, differential equations, and dynamics. I'll mention a few results that I'm aware of, giving links to literature surveys on different topics.
There is a short exact sequence $$Diff_0(M)\to Diff(M)\to MCG(M),$$ where $Diff_0(M)$ is the subgroup of diffeomorphisms isotopic to the identity.
One can regard $MCG(M)=\pi_0(Diff(M))$. There is a huge literature studying $MCG(M)$, especially when $M$ is a surface. One question that has been answered for closed surfaces is that there is no section $Diff(M)\leftarrow MCG(M)$. I'm not sure what's known about the higher-dimensional version of this question.
Topologists study the homotopy type of $Diff(M)$, which breaks down into computing $MCG(M)$ and the homotopy type of $Diff_0(M)$. Hatcher has a survey on the homotopy type of $Diff(M)$. This has more-or-less been completely resolved in dimensions $\leq 3$and $S^n$, , but is quite complex for general higher dimensional manifolds.
It is known that $Diff_0(M)$ is simple for closed manifolds by a result of Thurston. A general strategy then for understanding the group structure of $Diff_0(M)$ is to understand its subgroups. One aspect of this is the Zimmer program, to understand homomorphisms $\Lambda\to Diff_0(M)$, where $\Lambda$ is a higher rank lattice. Another aspect is to consider homomorphisms between diffeomorphism groups for different manifolds.
There are some results on dynamics of diffeomorphisms with relation to the diffeomorphism group. There is a huge literature on the dynamics of individual diffeomorphisms, but I think this is orthogonal to your question.
