Apollo 11

Apollo 11 landed on the Moon on July 20, 1969. Humans regarded this as the first Moon landing.

Motivations for the landing
Around 1959, a group of aliens had landed on the Moon and sent a transmission to Earth. They requested a meeting with Humans so as to share technology. The American government intercepted the transmission and started NASA's moon project to meet them. In the popular imagination, the Moon landing had much to do with the Americans' rivalry with the Soviet Union. (WC: Blue Moon)

The mission
Coordinated by Mission Control at the Kennedy Space Center, the Apollo 11, powered by a Saturn V rocket, lifted off on July 16. En Route to the Moon, the crew encountered unnamed aliens. Mission Control, as well as Torchwood, would know of this, but not the public. After getting locked by an overzealous secret services officer, Spencer, the Doctor ended up locked up in the Center, though gained release in enough time to greet the aliens due to land on the Moon in a flying saucer prior to the Apollo. The Doctor convinced the aliens that Humanity needed to develop on their own. The Doctor then witnessed the landing. (WC: Blue Moon)

Public reaction
A crowd gathered to watch the takeoff from the ground. (WC: Blue Moon) With interest in the landing so intense, public places virtually emptied as people all over the world glued themselves to their televisions. (ST: One Small Step) The Doctor and his companion Martha Jones, temporarily stuck in 1969, also watched the event of television. (DW: Blink)

Minor references and mentions
The Doctor and Martha had also personally witnessed the Moon landing four times together, as opposed to watching it on television. (DW: Blink)

Behind the Scenes
Prior to the real actual touchdown on the Moon, Moon Landing in TV Comic had already (obviously inaccurately) shown it has happening for the first time in 1970.