Can anyone suggest a good overview/introduction of the Floer machine for a beginner? (Someone pointed out some intriguing connections to surface mapping class groups, which might be enough incentive to dip the toe in (though I suppose "total immersion" might be the only realistic option...)
Michael Hutchings' lecture notes were precisely for this purpose; posted on his webpage: http://math.berkeley.edu/~hutching/
Lecture Notes on Morse Homology (With an Eye Towards Floer Theory and Pseudoholomorphic Curves)
Dietmar Salamon's notes are my favorite:
Also I should point out that the geometric intuition provided by Andreas Floer in some of his early papers is really quite beautiful and illuminating. For example read the introduction to his 1989 paper, Symplectic Fixed Points and Holomorphic Spheres, in Comm. Math. Phys (120) 575-611.
For the grittier details it's better to look in something like Dietmar's notes or the big book of J-holomorphic curves.
I wholeheartedly agree with both of Chris Gerig's suggestions.
The small McDuff-Salamon book on holomorphic curves ("J-holomorphic curves and quantum cohomology", available on McDuff's webpage) has a small chapter on Floer homology. The ideas in the rest of the book are also useful for Floer theory.
Audin and Damian have an introductory book called "Théorie de Morse et homologie de Floer". I haven't read it, but I hear good things about it.
A very good place is Kronheimer and Mrowka's monograph on monopole homology. Chapter 1 goes through the finite dimensional part.
I can vouch for Audin-Damian's Theorie de Morse et Homologie de Floer, read it cover to cover for my quals. They do Hamiltonian Floer theory with simplifying assumptions ($\omega$ and $c_1$ vanish on $\pi_2$ so there's no need to worry about bubbling, grading issues or caps, which one can learn from Dietmar's notes). They prove everything and provide intuition all along. The most technical estimates used for gluing are grouped into a Chapter that one can skip without loss of understanding. It also does Morse theory as a warm-up.