The entirety of the US vs USSR space race began on October 4th, 1957 with the launching of Sputnik. This Soviet satellite was attached to a R-7 ICBM, that also proved that the USSR was capable of a nuclear strike against the United States.  They went on to launch Sputnik II only a month later, with a dog to prove that a human can survive spaceflight. Although the United States was able to launch a satellite in 1958, they were once again outdone by the Russians in 1959 with the launch of Luna II. This was the first probe to hit the Moon. 
The world was shocked again in April of 1961 when cosmonaut Yuri Gagarin was the first to orbit the Earth. It took almost a year later, in February of 1962, for the American astronaut John Glenn to match this.
As an American, losing to the Soviets was not an option. From 1961-1964, NASA had a budget increase of 500% . John F. Kennedy’s speech in 1962 spurred increased spending and hiring by NASA. They were at a total force of about 400,000 people working on the space program.
In January of 1966, the United States capitalized (ha ha) on the death of the Soviets chief engineer for their space program. We were at full steam ahead and went on to power through the 60’s. In late 1968, Apollo 8 was the first manned mission around the moon.
The US officially won the space race on July 20th, 1969 when we launchedApollo 11. Astronauts Neil Armstrong, Buzz Aldrin made history by successfully landing on the moon after 4 days of travel. 
In conclusion, the United States won the Space Race. We were able to win against the USSR, and successfully beat them to the moon. Uncommonly known, they have four separate attempts and failures of landing on the moon.  After winning, the US space program was funded less and has not been what it was ever since.