You need to know what is the splitting of $f_* \mathscr{O}_X$, which depends on the cover.

Let us consider the double cover case, which is the easiest one. Then $$f_* \mathscr{O}_X = \mathscr{O}_Y \oplus L^{-1},$$ where $L$ is a line bundle on $Y$ such that $R \in |2L|$. Then, given any vector bundle $E$ on $Y$, by using projection formula one obtains $$F=f_*(f^*E)=f_*(f^*E \otimes \mathscr{O}_X) = E \otimes f_* \mathscr{O}_X = E \oplus (E \otimes L^{-1}),$$ so the computations of the Chern classes of $F$ is now a straightforward application of the splitting principle.

For general finite covers of degree $d$, one can only say that $$f_* \mathscr{O}_X = \mathscr{O}_Y \oplus V,$$ where $V$ is a vector bundle on $Y$ of rank $d-1$. As before, one obtains $$F=f_*(f^*E)=f_*(f^*E \otimes \mathscr{O}_X) = E \otimes f_* \mathscr{O}_X = E \oplus (E \otimes V),$$ but now the computations of Chern classes will be more difficult since in general $V$ would be indecomposable.

In some lucky cases, for instance when the cover $f \colon X \to Y$ is Galois with abelian Galois group, $V$ splits completely into line bundles and then, if one is able to identify $V$, the computation can be made again by using the splitting principle.

You need to know what is the splitting of $f_* \mathscr{O}_X$, which depends on the cover.

Let us consider the double cover case, which is the easiest one. Then $$f_* \mathscr{O}_X = \mathscr{O}_Y \oplus L^{-1},$$ where $L$ is a line bundle on $Y$ such that $R \in |2L|$. Then, given any vector bundle $E$ on $Y$, by using projection formula one obtains $$F=f_*(f^*E)=f_*(f^*E \otimes \mathscr{O}_X) = E \otimes f_* \mathscr{O}_X = E \oplus (E \otimes L^{-1}),$$ so the computation of the Chern classes of $F$ is now a straightforward application of the splitting principle.

For general finite covers of degree $d$, one can only say that $$f_* \mathscr{O}_X = \mathscr{O}_Y \oplus V,$$ where $V$ is a vector bundle on $Y$ of rank $d-1$. As before, one obtains $$F=f_*(f^*E)=f_*(f^*E \otimes \mathscr{O}_X) = E \otimes f_* \mathscr{O}_X = E \oplus (E \otimes V),$$ but now the computations of Chern classes will be more difficult since in general $V$ is indecomposable.

In some lucky cases, for instance when the cover $f \colon X \to Y$ is Galois with abelian Galois group, $V$ splits completely into line bundles and then, if one is able to identify $V$, the computation can be made again by using the splitting principle.