As far as I understood from my so far cursory look into a living review article by Poisson, Pound and Vega on The Motion of Point Particles in Curved Spacetime, it's a bit messy. But I think if you manage to go through GR, this should be manageable, as well. It will probably help if you've dealt with Green's functions before and even better if you've seen (some) renormalization techniques before.
Basically it boils down to the problem that already the path of a point particle is difficult to nail down. A quick look at the Coulomb force (and the associated energy term) tells you that even in flat space-time and a comoving observer - so basically just plain old electrostatics - you get divergences on the world line of the charged particle.
Taking more care to see where this goes then leads to equations of motions that don't look like the geodesics of the space-time you started with. This is where the idea "point particles don't follow geodesics" come from.
On page 21 they refer then to a redefinition of the space-time proposed by Detweiler and Whiting where the particle is included from the beginning and one rearrives at a description of its motion by a geodesic.
I happened to be at seminar on equations of motion in GR, two years ago - sadly, back then most of it went way over my head. But apparently, they finished the proceedings of the seminar (the link is right on the seminar webpage). Some (most? all?) of the chapters appeared on the arxiv. During the seminar, the organizers wanted this book to be a useful introduction to the topic. I didn't get a look at it yet, not even on the arxiv, but that's definitely also worth a look, specifically the chapter contributed by Pound, as he's also behind the LRR article. And the chapter contains the magical word "introduction"...