From what I gather from the era the next goals were pretty clear, even to the general public: "go to Mars" (or in the Soviet case "go to Venus"), and then go to Alpha Centauri.

If the Soviets would have won the race to the moon this might even have happened. But instead the Soviets decided to focus on space stations, and the US declared themselves winner and did largely nothing (by rejecting NASA's Space Transportation System proposal, which was also about a space station and a way to get there cheaply).