A simplified timeline summary of the Space Race between the Americans and Russians.
Russians
4th Oct 1957 Sputnik-1 first artificial satellite
3rd of Nov 1957 Laika the dog went into space
12th April 1961 Yuri Gagarin was the first Earth orbit
America
1958 -1963 Project Mercury 6 successful flights
1961 -1966 Project Gemini
1961-1972 Project Apollo put 12 astronauts on the moon
From this you can see that the Russians had a strong and early lead with some humiliating and disastrous moments for the US. But the Americans sort to level the playing field realising that they could not play catch up to the Russians so they simply picked a target that could not use the technology as either nation had it. That challenge was going to the moon and it needed a lot of money and good engineers – engineers were sourced from all over the place not just from within the US.
The Russians started well but then saw their efforts dissolve into failures along the levels of the Americans and so dove tailing the two nations space explorations stories.
These moments in history have inspired countless pieces of music, poetry, art and fiction and film.