Solar research today is essentially plasma physics. You need a solid foundation in understanding electrodynamics and how matter operates under the influence of magnetic fields. Beyond this, my best recommendation is to get into the highest-ranked graduate school you can that is consistent with what you can afford and how much financial support they are willing to provide you. I know this sounds like rather lame advice, but in solar physics, most of the apparently successful astronomers and physicists in today's competitive market come from leading schools.
The places I know about that are strong in solar research are: Harvard University, University of Chicago, Caltech, University of California, University of Arizona and I believe the University of Colorado. I am sure there are others. Your best bet is to go to your local library at your college, look up some articles on solar research, and see where the authors come from.