Let's look at the subject in more detail. First satellite, first human, first successful lunar mission -- clearly the Soviets.
First successful mission to another planet: Mariner 2 flyby of Venus, 1962, USA. Your listing of Venera 1 as the "first to reach another planet" neglects to mention that the spacecraft failed before it got there.
First successful mission to Mars: Mariner 4, 1964, USA.
First communications satellites: passive, Echo I, 1960, USA; active, Courier 1B, USA.
First spacecraft rendezvous in orbit: Geminis 6 and 7, 1965, USA.
First spacecraft docking in orbit: Gemini 8, 1966, USA.
First manned spacecraft beyond low earth orbit: Apollo 8, 1968, USA
First manned spacecraft in lunar orbit: Apollo 8, 1968, USA
First spacecraft to orbit another planet: Mariner 9, Nov 1971, USA
First mission beyond the inner solar system: Pioneer 10, 1973, USA
First flyby of Jupiter: Pioneer 10, 1973, USA
There are many others.
Now let's examine some of the Soviet space firsts:
First soft lander on the Moon: 3 Feb 1966, Luna 9, USSR, a success by any definition, sent back pictures, operated for 3 days on lunar surface
compared to:, Surveyor 1, first USA soft lander, landed 14 July 1966, operated for nearly 6 months on the lunar surface
First soft lander on Mars: Mars 3, Dec 1971, USSR, operated for 14.5 seconds on the surface, compared to Viking 1, first USA Mars lander, July 1976, operated for 6 years on the surface.
So the story that the USSR was the clear leader in early space exploration is clearly false. Both nations had impressive 'firsts', anyone who doesn't acknowledge the accomplishments of both has poor knowledge of the subject.