When I prepared this module, I didn't follow any particular textbook, but it turned out that most of the material can be found (even in the same order) in John F. Humphreys: A Course in Group Theory (Oxford University Press).
Some of the ideas for the lectures I got from Geoff Smith and Olga Tabachnikova: Topics in Group Theory (Springer). Even though this book is nicely written, it is not intended for a complete novice at Group Theory.
Two other books which I can recommend are Israel Grossman and Wilhelm Magnus: Groups and Their Graphs (MAA), which is mainly concerned with the graphical representation of finite groups, and M. A. Armstrong: Groups and Symmetry (Springer), in which for example a classification of the seventeen different Wallpaper Patterns can be found.
An introduction to Group Theory can also be found in almost every textbook on Algebra like for exmaple in David S. Dummit and Richard M. Foote: Abstract Algebra (Wiley).