As an undergraduate, I took a semester of point-set topology that used Munkres's book Topology, and we studied the fundamental group towards the end of the course. Following that, I took a semester of algebraic topology that used Greenberg and Harper's book Algebraic Topology: A First Course. Greenberg and Harper start off with homotopy theory and introduce higher homotopy groups. However, they don't go very far with homotopy theory before turning their attention to singular homology.
Although there are various things I don't like about Greenberg and Harper's book (for example, I didn't learn about simplicial homology until much later, and I think I would have understood singular homology better if I had first learned simplicial homology), I think that the approach of giving a brief introduction to homotopy groups before proceeding to homology theory works pretty well. It's good to emerge from a one-semester course at least knowing what higher homotopy groups are.