The answer is quite classical when $f \colon X \to Y$ is an unramified double cover.
In this case Riemann - Hurwitz formula gives
$g(X)-1 = 2g(Y)-2$.
Consider the following three natural maps:
$f^* \colon J(Y) \to J(X)$,
$Nm \colon \textrm{Pic}^0(X) \to \textrm{Pic}^0(Y), \quad Nm(\sum a_ip_i):= \sum a_if(p_i)$
$\tau \colon J(X) \to J(X)$,
where $f^*$ is induced by the pull-back of $0$-cycles, $Nm$ is the norm map and $\tau$ is the involution induced by the double cover $f$.
Then
$\textrm{Ker} \; f^*=\langle L \rangle$, where $L$ is a point of order $2$ in $J(Y)$;
the connected component of $Nm^{-1}(0)$ containing the identity coincides with the image of $I-\tau$. It is an Abelian subvariety of $\textrm{Pic}^0(X)$ of dimension $g(Y)-1$, that is denoted by $\textrm{Prym}(X, \tau)$.
Moreover, under the identification of $\textrm{Pic}^0(X)$ with $J(X)$, the principal polarization of $J(X)$ restricts to twice a principal polarization on $\textrm{Prym}(X, \tau)$.
The geometry of Prym varieties is very rich. In particular, Riemann-Hurwitz identity
$K_X =f^*K_Y$
induces subtle relations between the Theta divisor $\Theta$ of $X$ and the Theta divisor $\widetilde{\Theta}$ of $\textrm{Prym}(X, \tau)$.
You can look at [Arbarello-Cornalba-Griffiths-Harris, Geometry of algebraic curves, Appendix C] or at [Birkenhake-Lange, Complex Abelian Varieties, Chapter 12] for further details.
In the general case, it is possible to define the so-called generalized Prym varieties, at least where $f \colon X \to Y$ is a tame Galois branched cover. Look for instance at the paper of MERINDOL
"Varietés de Prym d'un revetement galoisien [Prym varieties of a Galois covering]"
Journal Reine Angew. Math. 461 (1995), 49-61.
Best Answer
Edit: As pointed out by Sasha, my original argument was not complete for case (2). One can argue as follows: suppose $C$ is a $(p,q)$ complete intersection, with $p\geq q$. The maximum number $\ell$ of points of $C$ lying on a line is $\leq p\ $ — otherwise the line is contained in $C$. By B. Basili, Indice de Clifford des intersections complètes de l’espace, Bull. S. M. F. 124, no. 1 (1996), p. 61-95, the gonality of $C$ is $pq-\ell$, hence $\geq p(q-1)$. But if $C$ is a 2-sheeted cover of a curve $D$ of genus $\leq 3$, a $g^{1}_3$ on $D$ lifts to a $g^1_6$ on $C$; similarly if $C$ is a 3-sheeted cover of an elliptic (or more generally hyperelliptic) curve $E$, a $g^1_2$ on $E$ lifts to a $g^1_6$ on $C$. Hence the gonality of $C$ is $\leq 6$, which is less than $p(q-1)$ if $q\geq 4$.