It's important to note that Luna 2 was an unmanned probe, and did not land or orbit the Moon. The first spacecraft to orbit the Moon was also Soviet, Luna 3, which flew past the Moon in October 1959 and sent back the first images of the lunar far side.
The first spacecraft to land on the Moon was the Soviet Union's Luna 9, which successfully soft-landed on February 3, 1966.
The first crewed spacecraft to orbit the Moon was Apollo 8, launched by the United States in December 1968. And the first humans to walk on the Moon were Neil Armstrong and Buzz Aldrin, who landed their lunar module Eagle on July 20, 1969, as part of the Apollo 11 mission.
