For Differential Geometry there's this wonderful book called "Analysis and Algebra on DIfferentiable Manifolds", which is a big problem book containing problems (and solutions).
For a self-studying student this is particularly helpful because these sorts of books provide us with a good feedback loop. If we attempt a problem, get stuck on and can't solve it, we can always read the first few lines of a solution to see a hint on where to go next, and if we are really stuck we can check the full solution.
Do such books exist for Algebraic Topology, or Differential Topology? Or is the best method to gain a similar sort of feedback loop to read the theorems and examples in a book like Hatcher or Guillemin and Pollack and attempt to prove the theorems and examples by hand?