Why didn't they follow the water in Planet of the Apes (1968)?

The short canon answer is that the lake was salt water and undrinkable. Though it's not stated in the movie, this would help to explain why they left what appeared to be a valuable water source. It takes water to survive so they had to find fresh water quickly in that hostile alien environment.

The longer version answer:

They came down into what appears to be a large high desert lake with another lake at about the 11:00 position as viewed through the pilot's window. The earth date according to the ship's monitor was reading Nov. 25, 3978.

As they scramble out of the sinking spaceship, the camera shows a partial panoramic view of what looks like a very large lake with no apparent outlet. The canyon they paddled their raft through was actually just a connecting channel between adjoining lake sections.

Later in the movie a still bandaged Taylor is trying to communicate with Zera and Cornelious about where he came from. They produce a map that shows the supposed 'lake' in the forbidden zone where his ship sank. As he traces his overland route for them you can clearly see that the lower western end of the lake narrows, twists and turns through a gorge until it eventually drain into the ocean.

If they would have taken that water route through to the ocean, they would have discovered no civilization and likely died of thirst. Luckily for them, the route they took was actually the shortest possible route to food, water and civilization.

I just found this production background data:

The opening scenes of the original Planet of the Apes (1968) use landscape to disorient the main characters, as well as the movie-watching audience, by showing the humans' spaceship landing in an unnatural-looking deep lake surrounded by red desert sandstone, leading everyone to believe the planet is an alien one.

Filming the opening of the movie in Lake Powell near Glen Canyon, Arizona, a paradoxical deep lake in a desert, unsettles those taking in the scene because it appears natural and yet exists only through artificial involvement (in actuality, the human intervention of damming the Colorado River).

The dry, tree-free and plant-free red rock environment of Lake Powell and the apes' planet is something unfamiliar to the majority of the Earth's population that dwells on the coasts. Having a desert environment suggests a habitat challenging to primate habitation. In retrospect, the desert represents the plant-free wasteland that the Earth has become. http://clarkesworldmagazine.com/blevins_12_11/