Talking about PDEs is way more complex and I didn't get into it. What do you mean by a good PDE solver? Generally a PDE solver is built by doing some spatial discretization and then either solving that with some numerical linear solver directly (if it's a time-independent problem, or if it's a fully-implicit discretization), or using an ODE solver. There are many different spatial discretization techniques, including finite element, finite difference, finite volume, and spectral methods. There are libraries for helping one discretize space using each of these methods.

But then there are "full-stop PDE solvers" which choose a specific PDE and solve that. For example, MATLAB and Mathematica can numerically solve diffusion-advection systems directly. And there are many user packages for different PDEs.

So to cover this fully, I would need to go through the available tools for numerical linear algebra and spatial discretization, but on the other hand cover some full-stop PDE solvers as well. This will be a different blog post. Without knowing your background or what you want to do, FEniCS is a good Python finite element toolbox that can then be used to build the linear equation or system of ODEs, from which you use other tools like those mentioned here to complete the solving. As for full-stop PDE solvers in Python, I am not sure and need to look into it myself...