Many of these are freely available online.
prerequisite reading:
Any textbook on introductory group theory/abstract algebra. For example: Niels Lauritzen, Concrete abstract algebra Chapter 2
Murray Elder, A short introduction to self-similar groups AustMS Gazette 2012
core reading:
Laurent Bartholdi, Growth of groups and wreath products an expanded version of a mini-course given at "Le Louverain", June 24-27, 2014.
Brian Bowditch, A course on geometric group theory, 2005
Marc Culler, Notes on free groups, 2005
Cornelia Druțu, Michael Kapovich, Geometric Group Theory
Steve Gersten, Introduction to hyperbolic and automatic groups, Notes from the Banff conference, 1996
Slava Grigorcuk and Igor Pak, Groups of intermediate growth: an introduction for beginners
Pierre de la Harpe, Topics in geometric group theory, The University of Chicago Press, 2000
Misha Kapovich, Lectures on geometric group theory, 2002
Clara Loh, Geometric Group Theory, 2017
Avinoam Mann, How Groups Grow
John Meier, Groups, graphs and trees : an introduction to the geometry of infinite groups, LMS Student Texts 73
Chuck Miller, Combinatorial group theory, Notes for the ANU Workshop January/February 1996
Volodymyr Nekrashevych, Self-similar groups
Walter Neumann and Mike Shaprio, A short course in geometric group theory, Notes for the ANU Workshop January/February 1996
Hamish Short (ed) MSRI notes on hyperbolic groups, (version pdf by public demand) Group Theory from a Geometrical Viewpoint (E. Ghys, A. Haefliger, A. Verjovsky, ed) Proc. ICTP Trieste 1990, WorldScientific, Singapore, 1991, 3--64
more advanced/specialised stuff:
Itai Benjamini and Oded Schramm, Percolation beyond Z^d, many questions and a few answers, 1996
Noel Brady, Tim Riley, Hamish Short, The geometry of the word problem for finitely generated groups, Advanced Courses in Mathematics CRM Barcelona, Birkhäuser-Verlag, 2007
Jim Cannon, Bill Floyd, Walter Parry, Introductory notes on Richard Thompson's groups, 1996
Etienne Ghys and Pierre de la Harpe, Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988), Progr. Math., 83, Birkhäuser Boston, Boston, MA, 1990. English translation
Bob Gilman, On the definition of word hyperbolic groups, 2002
Vagn Lundsgaard Hansen, Braids and Coverings: Selected Topics, volume 18 of London Mathematical Society Student Texts. Cambridge University Press, 1989
Alexei Myasnikov, Vladimir Shpilrain and Alexander Ushakov, Group-based cryptography, 2008
Vladimir Pestov, Hyperlinear and sofic groups: a brief guide, 2008
Vladimir Pestov and Aleksandra Kwaitkowska, An introduction to hyperlinear and sofic groups, 2009
Gábor Pete, Probability and geometry on groups, Fall 2009
Mark Sapir, Some group theory problems, arXiv 0704.2899
Mark Sapir, Asymptotic invariants, complexity of groups and related problems, arXiv 1012.1325
Henry Wilton's blog for a course on geometric group theory 2009
a short history of the word problem, mactutor st andrews