de la Harpe's book is quite nice and has an amazing bibliography, but it doesn't really prove any deep theorems (though it certainly discusses them!). Some other sources.
1) Bridson and Haefliger's book "Metric Spaces of Non-Positive Curvature". Very easy to read and covers a lot of ground.
2) Ghys and de la Harpe's book on hyperbolic groups. Another classic, but in French. If you look around the web, you can find English translations.
3) Cannon's survey "Geometric Group Theory" in the Handbook of Geometric Topology is very nice.
4) Bowditch's survey "A course on geometric group theory" is also very nice.
5) Bridson has written two beautiful surveys entitled "Non-Positive Curvature in Group Theory" and "The Geometry of the Word Problem". The latter was one of the first things I read in any depth.
6) Geoghegan's "Topological Methods in Group Theory" is very nice, with a more topological approach.
7) Mike Davis's "The Geometry and Topology of Coxeter Groups" is a bit specific, but covers a lot of important material in a nice way.
8) John Meier's book "Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups" is well-written and pretty gentle.
Best Answer
Analytic ideas enter into several parts of geometric group theory. Tom already mentioned amenability, so I'll skip that.
1) Complex analysis and the theory of quasiconformal mappings plays an important role in understanding the mapping class group, which is one of the most important groups studied by geometric group theorists. For information on this, see Farb and Margalit's forthcoming book "A primer on mapping class groups", available on either of their webpages.
2) The study of groups with property (T) ends up using quite a bit of analysis. See the book "Kazhdan's Property (T)" by de la Harpe, Bekka, and Valette for information on this.
3) Analysis (together with ideas from ergodic theory, which is of course quite analytic) plays an important role in proving various rigidity theorems. The most famous is the Mostow Rigidity Theorem, whose original proof uses lots of analysis : quasiconformal mapping in high dimensions, the fact that Lipschitz functions are differentiable almost everywhere, ergodic theory, etc.