Let $M$ by an compact, connected $n$dimensional manifold without boundary. Are there any other computable examples of the StiefelWhitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, \mathbb{H}P^m$?

3$\begingroup$ It is also not too hard to compute the sw classes of projectivizations of vector bundles over these manifolds $\endgroup$ – Thomas Rot Sep 21 '15 at 6:06
It goes back to Wu in the 1950's that if one can compute the mod 2 cohomology of a manifold, with its Steenrod operations, then one can explicitly compute its StiefelWhitney classes, via the Wu formula. See for example the Theorem on page 188 of ``A concise course on algebraic topology'' (no originality claimed, just the quickest reference for me to find).
The total StiefelWhitney class of any connected compact orientable surface $M$ can be computed fairly easily. As $M$ can be embedded in $\mathbb{R}^3$ with trivial normal bundle, $TM$ is stably trivial and therefore $w(M) = 1$. Alternatively, $w_1(M) = 0$ as $M$ is orientable and $w_2(M) = 0$ as it is the mod $2$ reduction of the Euler class which is $(2  2g)a$ where $a$ is the generator of $H^2(M, \mathbb{Z})$.
The manifold $P(m,n) := (S^m\times \mathbb{CP}^n)/(\mathbb Z/2)$, where $\mathbb Z/2$ acts by the antipodal map on $S^m$ and by complex conjugation on $\mathbb{CP}^n$, is called a Dold manifold. In "Erzeugende der Thomschen Algebra $\mathfrak N$", Dold computes the mod 2 cohomology, Steenrod module structure, and StiefelWhitney classes of $P(m,n)$. The calculations are fairly straightforward, and I found them to be a good example when I was learning StiefelWhitney classes and wanted more examples than projective spaces.
The paper is in German, but it's nonetheless quite readable, especially with a dictionary.