More specifically, what do we know about the following problem?
Given a smooth, compact manifold $M$ of dimension $n \geq 3$, does there exist a metric $g'$ conformal to $g$ for which the Ricci curvature of $g'$ is constant?
I've looked for this online but didn't find anything.
Apart from dimension 2, the answer to your question is negative. Take a look at the paper
A. Rod Gover, Pawel Nurowski, Obstructions to conformally Einstein metrics in $n$ dimensions, Journal of Geometry and Physics, Volume 56, Issue 3 (2006) pages 450-484.
for certain obstructions. Interestingly, under some genericity assumptions they proved that their obstructions are both necessary and sufficient conditions for a metric to be locally conformal to an Einstein metric. (The paper is sloppy about distinguishing local and global.) I am not sure if their conditions are also sufficient for a metric to be globally conformally Einstein. This looks like a good research project.
Few more remarks: A Riemannian metric on a connected 3-manifold is Einstein iff it has constant curvature. Thus, you get many 3-manifolds which do not admit Einstein metrics (say, $S^1\times S^2$). In dimension 4, say, a compact connected manifold of negative Euler characteristic does not admit an Einstein metric (an observation due to Berger). The existence of an Einstein metric on an arbitrary manifold of dimension $\ge 5$ is an important open problem.