Last Friday I mentioned some math that you'd probably run in to when trying to study climate. You'll also encounter some of the basic sciences (as opposed to messy sciences like climate).
Extremely common as requirements are:
College Chemistry (year long sequence)
Physics (year long calculus-based sequence)
Thermodynamics (from one or another of Chemistry, Physics, or Engineering departments)
Plus, though I've never seen it required, it seems quite common for people to take it:
Astronomy. I think this is more a matter of personality than requirement.
Many people arrive in climate by way of physics, so they'll also have a year of modern physics, intermediate mechanics, intermediate electricity and magnetism, and some physics lab courses.
If they go to climate through geology, they'll have the above extremely common basic science courses, but then a batch of different messy science (geology) courses.