Maybe the easiest non-trivial case to consider is when one has a double cover $f \colon X \to Y$ of smooth projective varieties. Such a double cover is defined by a smooth divisor $B$ in $Y$ which is $2$-divisible in the Picard group of $Y$, and by a choice of a square root $L$, namely $L^2 =B$. Then

$f_* \mathcal{O}_X=\mathcal{O}_Y \oplus L^{-1}$.

Now take a line bundle $\mathcal{O}_X(D)$ in $X$.

Then $f_{*} O_X(D)$ is a rank $2$ vector bundle on $Y$, and its Chern classes can be recovered in terms of $D$ and $L$ by using Grothendieck-Riemann-Roch. I think these computations were first carried out by Schwarzenberger in the case of smooth surfaces, see [Vector bundles on algebraic surfaces, Proc. London Math. Soc. 11 (1961)] and [Vector bundles on the projective plane, same journal].    

**Proposition**

Let $f \colon X \to Y$ and $L$ be as above, $D$ a divisor on $X$.

**(1)** We have the following equality in $\textrm{Pic}(Y)$:

$c_1$ 
$f_*$ 
$\mathcal{O}(D))$= $[f_* D]-L$.

**(2)** We have the following equality in $H^4(Y, \mathbb{Z}[1/2])$:

$c_2$ 
$f_*$ 
$\mathcal{O}(D))= 1/2((f_*D)^2-f_*(D^2)-f_*D \cdot L)$.