You obviously have all characteristic classes of $E$, of $F$, and of $E/F$,
with some relations because $E\cong F\oplus E/F$. To see if there are more, consider the classifying space for your pairs. If you are working over $\Bbbk$, $\mathrm{rk}(F)=k$, $\mathrm{rk}(E)=\ell$, it is the colimit over $n$ of the space of partial flags of length $(k,\ell)$ in $\Bbbk^n$. For a fixed $n$, this is homotopy equivalent to
$$G_{k,\ell,n}(\Bbbk)=U(n,\Bbbk)/(U(k,\Bbbk)\times U(\ell-k,\Bbbk)\times U(n-\ell,\Bbbk))\;.$$
This space is the total space of (at least) three fibre bundles $G_{k,\ell}(\Bbbk)\hookrightarrow G_{k,\ell,n}(\Bbbk)\twoheadrightarrow G_{\ell,n}(\Bbbk)$ and $G_{\ell-k,n-k}(\Bbbk)\hookrightarrow G_{k,\ell,n}(\Bbbk)\twoheadrightarrow G_{k,n}(\Bbbk)$ and $G_{k,n+k-\ell}(\Bbbk)\hookrightarrow G_{k,\ell,n}(\Bbbk)\twoheadrightarrow G_{\ell-k,n}(\Bbbk)$. Hence, you can try the Leray-Serre spectral sequence to find characteristic classes not generated by those mentioned above.

In the holomorphic setting, you see Bott-Chern forms. If $E$ and $F$ are flat, you see the forms described in the appendix of Bismut-Lott.
In both settings, you can sometimes extract cohomological information.

I am not sure there are similar classes in a purely topological setting, but I would personally go for $\Bbbk=\mathbb R$ first and look at $\mathbb Z/2$-valued cohomology.

EDIT: It seems there are no additional characteristic classes that do not come from $F$ or $E/F$. For consider the fibre bundle $G_{\ell-k,n-k}(\Bbbk)\hookrightarrow G_{k,\ell,n}(\Bbbk)\twoheadrightarrow G_{k,n}(\Bbbk)$. For sufficiently large $n$, all classes of $G_{\ell-k,n-k}(\Bbbk)$ of small degree come from the vector bundle $E/F$, and so are pulled back from $G_{k,\ell,n}(\Bbbk)$. Hence by the Leray-Hirsch theorem, in small degrees the cohomology ring of
$G_{k,\ell,n}(\Bbbk)$ is generated by the characteristic classes of the vector bundles $F$ and $E/F$.