For $a > 0$,
$\displaystyle P\{XY > a\} = \int_0^\infty \int_{\frac ax}^\infty f_{X,Y}(x,y)
\,\mathrm dy\,\mathrm dx + \int_{-\infty}^0 \int_{-\infty}^{\frac ax} f_{X,Y}(x,y)
\,\mathrm dy\,\mathrm dx$.
From this, you can deduce $P\{XY \leq a\} = F_{XY}(a)$ for $a >0$ and hence the
density function for $a >0$ by differentiating $F_{XY}(a)$ with respect
to $a$. A similar calculation can be done for $a < 0$; setting up
the integrals and their limits is left to you as an exercise.
If you understand
the Fundamental Theorem of Calculus well enough to be able to differentiate
an integral with respect to a parameter that appears only in the limits,
then you can even avoid calculating the inner integrals in both cases.
Some textbook writers even exhibit the pdf of $XY$ as a mystical magical
integral involving absolute values etc that gives the answer for all
$a \in (-\infty,\infty)$, but using this formula in any actual calculation
usually results
in disaster for beginners. I am not a trained professional by any means,
but I still advise you not to try it at home.
Note that this answers both questions asked as long as you know the
joint pdf
of $X$ and $Y$ (e.g. $X$ and $Y$ are independent with known pdfs) but
not if you know only that $X$ and $Y$ are correlated without knowing
the joint pdf. As Xi'an says in a comment, your second question
has no answer in general
