In addition to the references suggested in the above comments, I'd recommend taking a look at recent work of B. Khesin, D. Ebin, G. Misiolek, S. Preston, P. Michor among others... B. Khesin has a nice book
(freely available on his webpage) about infinite-dimensional groups, with a whole chapter on Diffeomorphism groups, that is perhaps a good place to start.

Just for a glimpse of what goes on beyond "just" topology, exotic structures, etc., the geometry of a few interesting subgroups of the diffeomorphism group $\mathcal D^s(M)$ of a manifold $M$ is also an important object of study. For example, take $\mathcal D_\mu^s(M)$, formed by *volume preserving* diffeomorphisms. This group is very much related with classical equations of hydrodynamics: you can think of the motion of an incompressible fluid filling a manifold $M$ as a curve in $\mathcal D_\mu^s(M)$. Classic work of Arnold (1966) and Ebin & Marsden (Ann. of Math., 1970), later followed the above mentioned authors, establishes a very beautiful setup for the Euler equations $\partial_t u+\nabla_u u=-\nabla p$, $div\ u=0$ (and something similar can be done for Navier-Stokes), proving that solutions $u(t,x)$ to these PDEs on $M$ are the $1$-parameter families of volume preserving diffeomorphisms that arise as geodesics in $\mathcal D_\mu^s(M)$ for an $L^2$ Riemannian metric in this infinite-dimensional manifold. This approach also works for some other PDEs, like Burgers' equation and KdV, the latter having to do with the diffeomorphism group of the circle. Most of this is discussed in detail in Khesin's book.

In this way, weak Riemannian geometry of the diffeomorphism group of $M$ and some of its submanifolds is deeply interconnected with many evolution equations on $M$. Of course, as the OP mentions, low dimension plays a big role in having more answers for now. For example, one can prove global existence of solutions to some of these PDEs of hydrodynamics in 2D using the above setup, but the 3D version is open and worth some big prizes and big money.

It is also worth pointing out that, recently, the infinite-dimensional geometry of diffeomorphism groups has been related to areas other than hydrodynamics, like optimal transport and geometric statistics, see this paper.

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