I'm doing my best to keep the math to a minimum here, because one can indeed understand a lot of the fundamentals without it, also because I know many people suffer from math anxiety. Then a large contribution from the fact that I know if I start down the road towards using math, I'll use a lot of it.
Now, on the math anxiety side, I highly recommend Sheila Tobias' Overcoming Math Anxiety. Unlike your probable expectations, she is not some outsider who never had the problem herself. She had a big case of it, but decided that physics was interesting and she'd bite the bullet and try to learn the math needed. She did, along the way dealing with a number of anxieties and false impressions about mathematics. One of the major problems shared by many people is the notion that only certain 'special' people can do mathematics. If you're talking the seriously hard core stuff that wins Fields Medals, that's probably true. If you're thinking about the level that I'm using here, the answer is ... nonsense. Anybody without a serious learning disability can learn this, regardless of age, gender, race, religion, part of world you live in, etc.
Here's a sampling of courses or material that one would almost certainly encounter on the way to being able to study climatology professionally:
Calculus of Several Variables
Probability and/or Statistics
Ordinary Differential Equations
That's just warming up. You'd also likely encounter several of:
Partial Differential Equations (probably multiple courses)
Numerical methods (probably multiple courses)
Statistics (in a multi-term sequence)
... and probably several more that aren't leaping to my mind right now.