I have been trying to learn about spherical harmonics since high school, however I have not been able to find a reference that lays out the concepts simply and clearly in such a way that it is easy to organize my further study into their properties.
Ideally I would like to be able to understand why spherical harmonics are the shapes of electron orbitals, without having to explore the theory in depth using advanced differential geometry which I don't know yet.
As an example of what I have in mind, consider this short paper about elliptic functions. It explains the motivation for the Jacobi elliptic functions in a way that a high school calculus student could understand. Algebraic Geometry: A Problem-Solving Approach by Garrity et al is also a good example of an introductory text that is able to first motivate the subject using basic mathematics, and then incrementally uses more techniques to show how each level of sophistication adds new insights that would have been difficult or impossible had we kept the math "easier".