Books for Hyperbolic Geometry.

A professor of mine suggested Euclidean and Non-Euclidean Geometries: Development and History by Marvin J. Greenberg when I asked him the same question.

Geometry and Topology of Three-Manifolds by Bill Thurston, edited by Silvio Levy.

There's 56-page introductory paper on hyperbolic geometry by Cannon, Floyd, Kenyon, and Parry which is absolutely excellent. You can download it from Kenyon's web page:

Cannon, Floyd, Kenyon, and Parry, Hyperbolic Geometry.

Note that this paper is much more directed towards the modern point of view than most sources, and is not at all interested in synthetic (or axiomatic) geometry.