The diffeomorphism groups $\text{Diff}(M)$ are sensitive to stabilization, say replacing $M$ by $M \times [0,1]$, so the direct contribution of the homotopy type of $M$ to $\text{Diff}(M)$ can be obscure. If you instead look at the concordance = pseudoisotopy spaces $$P(M) = \text{Diff}(M \times [0,1] \ \text{rel}\ M \times \{0\}),$$ then the stabilization maps $P(M) \to P(M \times [0,1])$ get highly connected as the dimension of $M$ grows (by Kiyoshi Igusa's stability theorem), hence the low-dimensional homotopy and (co-)homology of $P(M)$ agrees with that of the stable pseudoisotopy space $$\mathscr{P}(M) = \text{colim}_n P(M \times [0,1]^n).$$ The homotopy type of $M$, being the space of points in $M$, and the homotopy type of the free loop space $\mathscr{L}M = Map(S^1, M)$, being the space of closed loops in $M$, both contribute to $\mathscr{P}(M)$, basically through maps $$\mathscr{P}(*) \times M \to \mathscr{P}(M)$$ and $$\mathscr{P}(S^1) \times \mathscr{L}M \to \mathscr{P}(M).$$ See the paper

https://projecteuclid.org/download/pdf_1/euclid.jdg/1214447541

of Tom Farrell and Lowell Jones. There is a naturally defined involution on $P(M)$, and by the work of Allen Hatcher, Michael Weiss and Bruce Williams you can use it to largely recover $\text{Diff}(M)$ from $P(M)$. A more precise statement involves the block diffeomorphism group $\widetilde{\text{Diff}}(M)$, which is quite well understood by surgery theory. The survey "Automorphisms of manifolds" by Weiss and Williams might be a good source. By the stable parametrized $h$-cobordism theorem, written up by Friedhelm Waldhausen, Bjørn Jahren and myself, the spaces $\mathscr{P}(*)$ and $\mathscr{P}(S^1)$ are very close to Waldhausen's algebraic $K$-theory spaces $A(*)$ and $A(S^1)$, which agree with the algebraic $K$-theory spaces of the ring spectra $S$ and $S[\mathbb{Z}]$, respectively. I have some papers on $K(S)$, and Lars Hesselholt has more information about $K(S[\mathbb{Z}])$. I think this is one of the main reasons to be interested in the algebraic $K$-theory of ring spectra.