Just to get started (a bit long for a comment):

For a fiber bundle $F\rightarrow E\rightarrow^{\pi} B$ one has an exact sequence of bundles over $E$

$$ 0\rightarrow T_vE\rightarrow TE\rightarrow \pi^{*}TB\rightarrow 0. $$

Here $T_vE:=\ker T\pi$ is the vertical part of the tangent space to $E$, i.e. these are the tangent vectors to the fibers. The choice of a complementary bundle amounts to the choice of a connection, but can be identified with $\pi^*TB$. So one sees that

$$ w(TE)=w(T_vE)\pi^*(w(TB)). $$

I think one needs more information if one wants to continue the computation. For example one can go on if $E$ is the projectivization of a vector bundle over $B$.

Just to get started (a bit long for a comment):

For a fiber bundle $F\rightarrow E\overset{\pi}{\rightarrow}B$ one has an exact sequence of bundles over $E$

$$ 0\rightarrow T_vE\rightarrow TE\rightarrow \pi^{*}TB\rightarrow 0. $$

Here $T_vE:=\ker T\pi$ is the vertical part of the tangent space to $E$, i.e. these are the tangent vectors to the fibers. The choice of a complementary bundle amounts to the choice of a connection, but can be identified with $\pi^*TB$. So one sees that

$$ w(TE)=w(T_vE)\pi^*(w(TB)). $$

I think one needs more information if one wants to continue the computation. For example one can go on if $E$ is the projectivization of a vector bundle over $B$.