Enjoyable book to learn Topology.

Solution 1:

For general topology I would very much advise the book Topology by James R. Munkres.

I wouldn't be able to advise you very well for a book of algebraic topology, because I'm just learning the subject myself. Probably a good place where to start are Algebraic Topology by Hatcher (available online) and Topology and Geometry by Bredon.

Solution 2:

Czes Kosniowski's A First Course in Algebraic Topology changed my life.

Hatcher and Munkres are also wonderful, but in any case, it's likely that the answers you will receive will correspond to the answerers' respective introductions to the subject.