Background Material

If you need a refresher on how to do proofs:
Introduction to mathematical arguments, by Michael Hutchings.

Book of Proof, by Richard Hammack, contains an introduction to sets and a chapter on cardinalities.

Sets, arithmetic, and introduction to proofs are covered in MathDoctorBob's video course Math Major Basics.

Additional Resources

Other algebra texts: There are many that you can find in the library. A rather incomplete list: Michael Artin, Algebra, John Fraleigh, A First Course in Abstract Algebra, Joseph Gallian, Contemporary Abstract Algebra, Thomas Hungerford, Abstract Algebra: An Introduction, Serge Lang, Undergraduate Algebra.

More group theory videos:
Modern algebra, online course by Benedict Gross, includes video.
Group Theory Videos by Ladislau Fernandes.

Other online notes: For completeness, I've provided links to several other online course notes on group theory.
Group Theory Notes, by Donald L. Kreher.
Notes on Group Theory, by Mark Reeder.
Group theory by J.S.Milne.
Group Theory via Rubik's cube by Tom Davis.

Advanced: A partial list of more advanced and fast-paced online books on group theory and algebra:
Paul Selick's algebra course.
Abstract Algebra by Robert B. Ash.
Abstract Algebra by Paul Garrett.
Notes on Algebra, by Marc Culler Section on group theory.
Class notes for modern algebra by Matt Douglass