Three paragraphs seem quite limited to cover three decades of the space race, but for any length research paper on this topic, NASA's website is certainly a good starting point. You'll find a lot of helpful information on the launch of the space age/race and more than enough information for your short paper.

The former USSR's launch of Sputnik kicked in a sense of national pride and demonstrated our disdain for Russia at that time.  We thought we were the best in everything and seeing a nation we were in a Cold War with beat us into space was a blow to our national pride.  It also prompted fears of being invaded from space by other nations.  These were the days when we had "Air Raid drills"  in school.

Along with many successes in the "space race," we have had many challenges and some disasters.  You may want to mention notable "firsts" in our exploration of space, but be sure to mention those unsuccessful ventures as well so your paper is well-rounded and accurate.

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.