I don't think that trying to solve the Schrödinger equation itself is particularly useful. The SE is a partial differential equation, and there's a whole logic of differential equations and boundary conditions, etc. that provides context for the SE. If you're serious about trying to understand quantum mechanics, I think the concept of Hilbert space/abstract vector spaces/linear algebra in general is a bigger conceptual shift than just being able to solve the particle in a box in function space. It's also just a really useful set of concepts that makes learning things like optimization, coordinate/fourier transforms, etc. easier/more intuitive.

Until I had the wave function explained to me as some vector in a high dimensional space that we could map into x-space or p-space or Lz-space I don't think I really had a good grasp on quantum mechanics. This is anecdote not data, your mileage may vary.

Try the wave equation first? If you want to think of particles like waves it might be useful to know what a wave is. Note that you already need to have heard of a respectable chunk of calculus to solve this equation.

The Schrodinger equation itself can be understood as simply a particular instance of integral calculus. If I recall my undergrad days correctly, you didn't even need linear algebra. Once you know the calculus, quantum waveforms don't require a whole lot of additional mathematical insight.