I think you could do a lot worse than to focus on modern applications to data analysis by using the texts of Edelsbrunner and Harer and/or Zomorodian as touchstones and an avenue towards current work in topological data analysis. These books are self-contained treatments that focus on Morse theory and homology over $\mathbb{Z}/2\mathbb{Z}$, and there are a lot of materials and software available for a course to be built from.
I think you could do a lot worse than to focus on modern applications to data analysis by using the texts of Edelsbrunner and Harer and/or Zomorodian as touchstones and an avenue towards current work in topological data analysis. These books are self-contained treatments that focus on Morse theory and homology over $\mathbb{Z}/2\mathbb{Z}$, and there are a lot of materials and software available for a course to be built from.