I don't see why I should not recommend my own book "Topology and groupoids" as a text on general topology from a geometric viewpoint and on 1-dimensional homotopy theory from the modern view of groupoids. This allows for a form of the van Kampen theorem with many base points, chosen according to the geometry of the situation, from which one can deduce the fundamental group of the circle, a gap in traditional accounts; also I feel it makes the theory of covering spaces easier to follow since a covering map of spaces is modelled by a covering morphism of groupoids. A further bonus is that there is a theorem on the fundamental groupoid of an orbit space by a discontinuous action of a group, not to be found in any other text.
The book has been adopted for a course at Harvard, and more details are available at
It is available from amazon at $31.99 and an e-version with hyperref and some colour is available at £5 from http://www.kagi.com.
The book has no homology theory, so it contains only one initial part of algebraic topology.
BUT, another part of algebraic topology is in the new jointly authored book "Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids"published in 2011 by the European Mathematical Society. The print version is not cheap, but seems to me good value for 703 pages, and a pdf is available on my web site. There is a thorough presentation of the history and intuitions, and it should be seen as a sequel to "Topology and groupoids", to which it refers often.
It gives a quite different approach to the border between homotopy and homology, in which there is little singular homology, and no simplicial approximation. Instead, it gives a Higher Homotopy Seifert-van Kampen Theorem, which yields directly results on relative homotopy groups, including nonabelian ones in dimension 2 (!!), and including generalisations of the Relative Hurewicz Theorem. Part I, up to p. 204, is almost entirely on dimension 1 and 2, with lots of figures. You'll find little, if any, of the results on crossed modules in other algebraic topology texts.
Will this take on? The next 20 years may tell!