
I was four years old at the time, and it's one of my earliest memories. On that summer night I remember standing in the front yard with a toy pair of binoculars in my hands, staring up at the Moon. The Moon was full that night, or nearly so; and I remember the excitement that gripped the country that day. You might think that for a four-year-old child everything would seem novel and exciting, but even I understood that something unprecedented was taking place.
My parents let me stay up to watch the first moon walk. It happened around 10 pm Central time, and I recall being distinctly disappointed by the TV images. It was difficult to see what was going on, and in my four-year-old way I supposed the camera was filming from the other side of the LEM, underneath the ship, so that only the astronaut's feet were visible on the other side (in fact the contrast between light and shadow was so stark, and the TV image so grainy, that only the lower portion of the LEM ladder was clearly illuminated).
I watched all the subsequent missions. As a pre-schooler I had the luxury of time, so I watched the Saturn V liftoffs (they were almost all launched around 9 am, with the exception of the last one, which I remember had a spectacular nighttime launch). I feel fortunate now that I have those memories.
One thing that's been missed in all the 40th anniversary hubbub is the fact that one movie in particular actually helped sell the idea of space travel to the public. That movie was George Pal's Destination Moon (1950).
Fanciful movies about travel to the Moon had been made, of course, such as Melies' A Trip To the Moon and the Korda brothers' Things To Come, but those movies didn't take the idea seriously.
Destination Moon was dead serious. The film was written by Robert A. Heinlein and tried to tell an exciting tale that hewed to absolute accuracy in the scientific details. No bug-eyed aliens, no meteor showers, no cat women of the Moon. 
Buck Rogers it wasn't, but would ticket buyers swallow the conceit that this nutty idea was feasible? So unfamiliar was the idea of space travel to the moviegoing public of 1950 that a couple of odd concessions were made by the filmmakers. First, a Woody Woodpecker cartoon was integrated, designed to explain to potential investors of the lunar project (and by extension the audience) the basic principles of space flight.
In addition, a surrogate for the audience was included in the crew, a simple-minded Brooklynite named Joe, added to the ship's company as a last-minute replacement. So naive is Joe that even when suiting up for a space walk he doesn't understand that there isn't any air outside the ship ("There's plenty of room for it", he reasons) and it's difficult for a viewer today to grasp that most Americans in that situation would have said the same thing in 1950.
In terms of story, Destination Moon manages to be both dull and melodramatic. In terms of the science, it gets the details wrong but is actually quite accurate in conveying the basics of what a lunar excursion would be like. And in fact, a fair number of wide-eyed kids in the movie theaters of America, witnessing the launch and triumphant return of the Luna in 1950, grew up to become the aeronautical engineers who built the rockets that went to the Moon. Because of that influence, Destination Moon is included in NASA's official timeline of the history of space travel.
Nineteen years after the movie was released, Robert A. Heinlein was at Kennedy Space Center to witness the launch of Apollo 11. You really have to wonder what he thought of it all.