Okay, I would say Sputnik is how it started, so that's good.
But then you should talk about people in orbit -- Yuri Gagarin and John Glenn, for example.
And then you need to talk about JFK and his challenge that the US should get a man on the moon within the decade in which he was speaking. That decade was the '60s, of course, and the US got men on the moon by July of 1969. That really has to be in there.
After that, you could talk about the space shuttles and the space station and how the space race got more cooperative.
You could talk a bit if you like about how China has just put a man in orbit and is thinking about going to the moon.