I have been taught that the motion of a two-body system can take the
form of an ellipse, a parabola or a hyperbola, but how can I derive
these trajectories explicitly in polar and eventually cartesian
coordinates?

Henceforth is r the distance between the bodies, M and m the
respective masses, etc.

mu*r'' = mu*r*(theta')^2 + G*M*m/r^2 ; mu := M*m/(M+m)

mu*r'' - mu*r*(theta')^2 - G*M*m/r^2 = 0 ; L := mu*r^2*r'

mu*r'' - L^2/(mu*r^3) - G*M*m/r^2 = 0 ; *r'

mu*r''*r' - L^2*r'/(mu*r^3) - G*M*m*r'/r^2 = 0 ; integrate

mu*(r')^2/2 + L^2/(2*mu*r^2) + G*M*m/r = E

Even if I solve the last DE, then im stuck with an inverse function.
The problem is analogous if I try to solve for the angle theta. Is
there a way to obtain r(t) and theta(t), that is a parametrization of
the trajectory in polar coordinates? The rest would follow
immediately. Thanks for any help =)