Adult swimming instruction could probably found either from the YMCA, local Red Cross chapter, local Boy Scout troop, community college/college, local lifeguard or, in some extreme cases, a dive shop.
The first thing required is confidence in yourself and knowing your own physical limits. Most non-swimmers freak when they get water up their nose or mouth. Practice breathing techniques. Then learn basic floating techniques because if you get into trouble, you can at least float. Afterwards, there are basic strokes to learn: The elementary backstroke, the breaststroke and sidestroke. Practice, practice, practice the basic strokes and floating in chest deep water. One can swim quite a distance using the elementary backstroke while expending a minimal amount of energy.
<edit>edited for clarity</edit>