If I were to ask you, "Who were the first person to walk on the Moon?", the answer would (or should) readily come to mind - the American astronaut Neil Armstrong. The date of his landing - July 20, 1969 - might even come to mind, as well. And certainly his quote: "That's one small step for man; one giant leap for mankind."
But Armstrong is one of many important historical spacefarers. In this answer, I'll look to go beyond Armstrong a little bit, and tease out some other memorable "firsts" in the world's history of exploring space. I'll be relying heavily upon the excellent "Windows to the Universe", an online database created by the University of Michigan's University Corporation for Atmospheric Research.
Steps for Man (and Woman)
First human in space: Yuri Gagarin, a Soviet cosmonaut, became the first human to enter space on April 12th, 1961. He traveled on the Soviet mission Vostok 1 ("Orient 1" / "East 1") and made one complete orbit of the Earth. On this mission, Gagarin spent one hour and forty-eight minutes in space.
First American in space: Alan Shepard became the first American astronaut to enter space on May 5th, 1961. He traveled on the American mission Mercury 3, and, unlike Gagarin, did not reach proper orbit height; Shepard only went up 116 miles. Shepard spent fifteen minutes in space.
First woman in space: Valentina Tereshkova, a Soviet cosmonaut, became the first woman to enter space on June 16th, 1963. She traveled on the Soviet mission Volstok 6 and spent three days orbiting the Earth. Notably, Tereshkova made a television broadcast back to the Soviet Union during the course of her time in space.
Other nationalities' "firsts":
First space crew: The Soviet Union launched the first space crew - cosmonauts Vladimir Komarov, Konstantin Feoktistov and Boriz Yegorov - on October 12th, 1964. These cosmonauts were part of the Soviet mission Voskhod 1 ("Ascent 1" / "Dawn 1"). None of the three cosmonauts wore space suits, and all three suffered from "space sickness" while on their mission.
First spacewalk: On March 18th, 1965, Soviet cosmonaut Alexei Leonov became the first man to walk in space as a member of the Voskhod 2 mission. His "space jaunt" certainly was not lengthy; it lasted all of ten minutes.
First computerized spacecraft: The "Molly Brown", which launched America's Gemini-Titan space program on March 23rd, 1965, was the first spacecraft "to carry a computer for guidance". Its computerized nature aided its ability to maneuver in space.
First live television broadcast from space: The American team of Apollo 7 - Walter Schirra, Donn Eisele and Walt Cunningham - was the first to produce a live television broadcast from space. They performed this feat on October 11th, 1968.
First American laboratory in space: "Skylab" earned this honor on May 25th, 1973. A team of three American astronauts then spent a little under a month on the lab, performing "experiments and observations".
First successful Soviet space station: Salyut 3 ("Salute 3") became the first working Soviet space station - previous attempts had failed - on July 3rd, 1974. For their first mission, two Soviet cosmonauts spent two weeks aboard the space station.
First use of the Space Shuttle: The first Space Shuttle - also known as the "winged, reusable spacecraft" - was launched on April 12th, 1981. The Shuttle was named "Columbia". In total, it went on four journeys into space.
And Mankind's Leap
First human on the Moon: As previously stated, Neil Armstrong was the American astronaut who earned this title on July 20th, 1969. He was working as a member of the American space mission "Apollo 11". Armstrong and fellow astronaut Edwin "Buzz" Aldrin spent a total of twenty hours - two spent "moonwalking" - on the Moon.
First use of the lunar rover: On July 26th, 1971, American astronauts David Scott and James Irwin became the first to successfully use the lunar rover (formally known as the "Lunar Roving Vehicle"). Scott and Irwin were members of the American mission Apollo 15.
Sources: