For an authentic experience (and cheaper than the
well-decorated cafe/restaurants by Haad Rin beach),
look at the more modest cafes where you might see
some Thais eating.
The best area for authentic cheap eats is defintely
Thong Sala, the main town on the island.
Me'n'u located 3 km outside of Thongsala on the
road to Haad Yao at Hin Kong offers high end fine
dining in a sublime tranquil location.
Drink
Sheesha in Chaloklam offers a classy take on the
Thai bar, and is defintely worth a look.