The ending of Rise of the Planet of the Apes is a farewell between Caesar and Will, and, in my opinion, it is the most important choice in the film, because it turns a rescue story into the start of a species leaving its keepers.
After the fight on the Golden Gate Bridge, the apes cross to the forest and gain their freedom. Will finds Caesar and asks him to return home. Caesar answers that his home is now with the apes in the woods. Will accepts this and leaves him there.
Why does this land? It could be argued that Caesar was never a pet who rebelled, but a member of the family who learned that the family was not enough. The shelter showed him cruelty, so the redwood forest is the first place that belongs to him. The scene also fits the film's structure, since Will's research set the uprising in motion and Will is the one who watches it end.
There is a cost on the human side too. Charles dies after taking the stronger ALZ-113 and Franklin is infected during testing, so Will's research costs human lives as well as freeing the apes. The credits make the larger cost explicit, setting up the human decline and ape rise that follow in the series.
For what happens next, dawn of the planet of the apes opens ten years after a virus wiped out most of humanity, with a colony of intelligent apes led by Caesar living in the forest outside San Francisco. That is the direct answer to the ending, and it is why the chronological order of the planet of the apes timeline starts here.
One small observation: the name Dodge Landon combines the names of two astronauts in the 1968 film, Landon and Dodge, and perhaps the shelter owner is a quiet nod to it. That is a reading, not a confirmed intention.