Hmmm.....the US was the first to send men behind the moon, first photograph of the whole earth from space, first to land on the moon, first to send a fly by past Mars and Mercury, first to Jupiter, first to Saturn, first to Uranus, first to Neptune, first to send spacecraft into interstellar space, first to land a probe on Mars, first to photograph an asteroid, first to impact an asteroid, first space shuttle, first manned space station (anyone claiming Salyut 1 for this honor must think a woman is no longer a virgin after the first kiss. The crew couldn't get in the door of Salyut 1), first returned moon rock and soil samples, oldest operational spacecraft (Pioneer 6), first car on the moon, first traverse of the asteroid belt.
Russia has, first man in orbit, first crew dead in flight, turtles to the moon, their Venera probes were impressive, including the first strength test of a titanium lenscap on Venus, first to kill a spacecraft with Phobos.
Look, the Russians have many accomplishments, but no honest measure of the Space Race, which concluded with the landing of Armstrong, Collins, and Aldrin on the carrier deck, can conclude anything but that the Russians came in second. Races are determined by who crosses the finish line first, people.