Let $S$ be given by
$$S= \left[(x,y,z) \in \Bbb{R}\;|\; x^2+y^2+z^2+xy+xz+yz=\frac12 \right]$$
and $$\omega = xdy \wedge dz\, -\, \frac {2z}{y^3} \, dx\wedge dy \,+\, \frac1{y^2}dz\wedge dx $$
Explain how to give an orientation for $S$ and compute $\int_S \omega$ with respect to that orientation.
I know it involves Stokes's theorem and that $d\omega = dx \wedge dy \wedge dz$ but I'm kind lost about the orientation and actually computing it.
If in fact $\mathrm{d}\omega = dx \wedge dy \wedge dz$ then by Stokes' Theorem, you are computing the volume of the region enclosed by $S$. This region happens to be an ellipsoid, tilted with respect to the given axes. You can figure out how to obtain the dimensions and orientation of the axes here. It turns out that this ellipsoid has semiaxes of length $1$, $1$, and $1/2$, so the volume of the ellipsoid is $2\pi/3$, and the value of the integral is $(2\pi/3) dx \wedge dy \wedge dz$.
That said, I am not so sure that $\mathrm{d}\omega = dx \wedge dy \wedge dz$ as you state, because the 2-forms are in cyclic order and will not change sign upon differentiation. In that case, the terms with $y$ will not cancel. The resulting integral, however, would be a mess and if this is homework, I'd be surprised if that was the intention.