As discussed in the course outline, the textbook for the course will be
There are many other books on analysis. As Pugh is missing some topics that we intend to cover, we'll supplement at points with Serge Lang's Undergraduate analysis. Other textbooks that you might find helpful are Walter Rudin's Principles of mathematical analysis (at a similar level to Pugh, a bit more precise but also worse with explaining intuition), or Protter and Morrey's Intermediate calculus (more basic than Pugh).
You might also like early sections of Calculus on manifolds by Michael Spivak, or the Multivariable calculus notes on Jerry Shurman's webpage for the material on differentiation and integration. If you want to find additional calculation problems, you might look at the later sections of a lower-level Calculus book such as Thomas' Calculus (but such a book has very little in the way of theory or proofs).
The Pugh book is very strong on the metric space material. You can find a somewhat less sophisticated viewpoint in the books of Ross, Abbott, or Lebl that you may have from earlier classes. You can find a more sophisticated viewpoint in the early sections of An introduction to topology and homotopy by Allan Sieradski.
Although it is beyond the scope of the course, bioinformatics majors particularly may be interested in Part III of Mathematical Models in Biology by Leah Edelstein-Keshet, which uses material from Analysis III to model biological systems.
We can have a lively online discussion in the UP e-classroom system. Asking and answering questions of your peers will be of great benefit to your development.
Wikipedia has a number of articles on mathematical topics. They are mostly well-written and accurate, and you should find many of the articles relevant to the content of ANA III to be fairly readable.