I was able to solve part (a) using spherical coordinates. Part (b) is a completely different story. I'm not sure how the bounds were derived from the corresponding integral. Would someone on MSE be kind enough to explain to me where the bounds come from each of the corresponding integrals? For example, I get the z-bounds but why was z solved for from the equation $x^2 + y^2 + z^2 = 1$ rather than $x^2 + y^2 = z^2$? Honestly, I'm very confused by this, so any help would be appreciated.
In general, we get the marginal PDF by integrating the joint PDF over the other variables. So to get $f_{X,Y}(x,y)$ we need to integrate $f_{X,Y,Z}(x,y,z)$ over $z.$ The support of $f_{X,Y,Z}(x,y,z) = \frac{3}{4\pi}$ is the solid sphere $x^2+y^2+z^2 < 1,$ so we need to integrate over all $z$ satisfying that bound. This is $$ -\sqrt{1-x^2+y^2} < z < \sqrt{1-x^2+y^2}.$$
Similarly, if we want $f_{Z}(z)$ we need to integrate over the region of the $x$-$y$ plane satisfying $x^2+y^2+z^2<1,$ i.e. over the sphere with radius $\sqrt{1-z^2}.$