I am now a first year graduate student and I am interested in algebraic topology for many reasons. I took an undergraduate course in metric topology, but I didn't take one in general topology. However, I have a more than sufficient background in abstract algebra, I will very soon take a course in differential geometry (I already took one as an undergraduate, but it was of an elementary flavor - now I am going to study manifolds, etc.) and I have (intensely) studied topics like measure theory, so maybe I am "too mature" to go through a textbook like Munkres'?
I think I have two choices: go deep into topology and study from a textbook like Munkres', or start head-on with an introductory algebraic topology textbook which is tolerant to people who never seriously studied general topology (example: Lee's Introduction to Topological Manifolds). My main two concerns are: the lack of enough time and the possibility of wasting some.
What do you think should be done? Do you recommend an alternative to the above two plans?
I also don't have advisers or anyone in my uni to talk to about my situation, so I would really appreciate your advice as I don't know where else I can find one.