Only include headers in another header if absolutely necessary. If the header can go solely in the source file then that's the best place. You can use forward declarations of classes in the header if you are only using pointers and references to them. Your DrawScene, GameState, Grid and InputHandler classes looks like they might fall into this category.
Note that C++ as a language does not care about the distinction between headers and source files. That's just an extremely common system used by developers to maintain their code. The obvious advantage of using headers is to avoid code duplication and helps, to an extent, to enforce the one-definition-rule for classes, templates and inline functions.