Check the paper Antonelli, P. L.; Burghelea, D.; Kahn, P. J.
*The non-finite homotopy type of some diffeomorphism groups.* 
Topology 11 (1972), 1–49. 

There  they mention an old result of Palais (*Homotopv theory of infinite dimensional manifolds*. Topology 5 (1966),  1-16) which states that  the identity component of    ${\rm Diff}_0(M)$ has the homotopy type of a *countable* $CW$-complex.

In their paper, Antonelli,Burghelea and  Kahn  prove that for many smooth manifolds (including spheres of dimension $\geq 7$)  the group ${\rm Diff}_0(M)$ does not have the homotopy type of a *finite* $CW$-complex. (This is highly nontrivial.)  

Above, by diffeomorphisms  they mean smooth diffeomorphism and the topology is the $C^\infty$-topology.