This is really difficult to learn, just master the basic methods, such as separation of variables, characteristic line method, integral transformation method, and so on. You can read the book first, read all these methods (preferably with examples), and find out the key steps and processes that you still don't understand. Sometimes the derivation of these methods requires some in-depth mathematical knowledge. If you don't understand, don't think about it. Just focus on the application and remember the method.

Remember the method and do more exercises. The theoretical parts or examples in general books can't cover all kinds of problems (for example, wave equations have three boundary conditions, and books generally only take Dirichlet boundary conditions as an example to illustrate the solution process. This is a problem involving two other boundary conditions in the exercise, so you need to try it yourself, and the method is similar to that in the book), so that you can master this method well.