I rather like Spanier, but have not read too much of it. Bott recommended it to us in 1964, but that was about all there was out there that was fairly comprehensive. Like Spanier, my later teacher Ed Brown Jr. started us off with simplicial complexes, which I think is a good way to get a hands on feel for algebraic topology. Notice most of us still use some triangulation reasoning when trying to give an elementary calculation or explanation. E.g. the double cover of projective space by the sphere, plus a triangulation, gives the fact that the euler characteristic of projective space is 1 in even dimensions. One also gets quickly the very useful Hurwitz formula for euler characteristics of branched covers of riemann surfaces.