For point-set topology the standard recommendation is Munkres, which I generally like.

I would preface any textbook on topology with the first chapter of Ishan's "Differential geometry". It builds the reason for studying topology and why the axioms have the shape they have in a wonderful crescendo, and at the end even dabs a bit into nets (non point-set topology). It's very clear and builds a lot of intuition.

Also, as a side dish in a topology lunch, the peculiar "Counterexamples in topology".