The homotopy-type of the group of diffeomorphisms of a manifold are fairly well understood in dimensions $1$, $2$ and $3$. For a sketch of what's known see Hatcher's "Linearization in three-dimensional topology," in: Proc. Int. Congress of. Math., Helsinki, Vol. I (1978), pp. 463-468.

Similarly, the finite subgroups of $Diff(M)$ are well understood in dimensions $3$ and lower. Hatcher's paper is a good reference for that as well, when combined with a few semi-recent theorems.

If you're interested in general subgroups of $Diff(M)$, there's still a fair bit of discussion going on just for subgroups of $Diff(S^1)$, as it contains a pretty rich collection of subgroups.

In high dimensions there's not much known. For example, nobody knows if $Diff(S^4)$ has any more than two path-components. See for example this little blurb. Some of the rational homotopy groups of $Diff(S^n)$ are known for $n$ large enough.

I wrote a survey on what's known about the spaces $Diff(S^n)$, and spaces of smooth embeddings of one sphere in another $Emb(S^j,S^n)$ a few years ago, here.

Getting back to your earlier question, groups of diffeomorphisms of connect-sums can be pretty compicated objects. In dimension $2$ it's already interesting. For example, $Diff(S^1 \times S^1)$ has the homotopy-type of $S^1 \times S^1 \times GL_2(\mathbb Z)$. Diff of a connect-sum of $g$ copies of $S^1 \times S^1$ has the homotopy-type of a discrete group provided $g>1$, this is called the mapping class group of a surface of genus $g$. It's a pretty complicated and heavily-studied object. In the genus $g=2$ case this group is fairly similar to the braid group on $6$ strands.

In dimension $3$, it's an old theorem of Hatcher's that $Diff(S^1 \times S^2)$ doesn't have the homotopy-type of a finite-dimensional CW-complex, as it has the homotopy-type of $O_2 \times O_3 \times \Omega SO_3$. I've been spending a lot of time recently, studying the homotopy-type of $Diff(M)$ when $M$ is the complement of a knot in $S^3$, and knot complements in general. The paper of mine I linked to goes into some detail on this.

From the perspective of differential geometry, the homotopy-type of $Diff(S^n)$ is rather interesting as it's closely related to the homotopy-type of the space of "round Riemann metrics" on $S^n$. This is a classic construction, is outlined in my paper but it goes like this: $Diff(S^n)$ has the homotopy type of a product $O_{n+1} \times Diff(D^n)$ where the diffeomorphisms of $D^n$ are required to be the identity on the boundary -- this is a local linearization argument. $Diff(D^n)$ has the homotopy-type of the space of round metrics on $S^n$. The idea is that any two round metrics are related by a diffeomorphism of $S^n$. So $Diff(S^n)$ acts transitively on the space of round metrics (with a fixed volume, say), and the stabilizer of a round metric is $O_{n+1}$ basically by the definition of a round metrics. Kind of silly but fundamental.