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...)

You might have a look at Andrew CottonClay's thesis. math.harvard.edu/~acotton – Ian Agol Apr 28 '12 at 2:59

Although it's not primarily about Floer theory, OzbagciStipsicz "Surgery on contact 3manifolds" certainly deals with some such connections (where Floer means HeegaardFloer or SeibergWitten in this context). In particular, the relationship between mapping class groups and Lefschetz fibrations is a particularly fruitful source of theorems (see Section 15.3). – Jonny Evans Apr 28 '12 at 7:40

I enjoyed this video of a talk by Dusa McDuff: msri.org/web/msri/onlinevideos//video/showVideo/3995 – Omar AntolínCamarena Apr 28 '12 at 15:06

1I like the book written by Krohemer&Mrowka Monopoles and 3manifolds – Siqi He Sep 16 '12 at 2:23
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)

2There is also Lectures on Floer Homology by Dietmar Salamon (particularly for symplectic Floer theory). – Chris Gerig Apr 28 '12 at 3:14
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) 575611.
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104177909
For the grittier details it's better to look in something like Dietmar's notes or the big book of Jholomorphic curves.
I wholeheartedly agree with both of Chris Gerig's suggestions.
The small McDuffSalamon book on holomorphic curves ("Jholomorphic 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.
If you're going for total immersion, a good place to start is with Seidel's early papers (e.g. arXiv:math/0105186, arXiv:math/9803083, arXiv:math/0309012) where you learn by watching him do things.

Audin and Damian is available in english now too. It is very clearly written. – user100272 May 16 '17 at 16:57
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 AudinDamian'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 warmup.