Space exploration began centuries ago if you take the telescope as the standard of measurement. It began thousands of years ago, if you base it on man’s vision as he explored the heavens and looked to the stars for guiding ships and to the clouds for the release of rain from the heavens.
Space travel is a different question and one that can be answered with more precise answers. The Soviet Union placed the first satellite, Sputnik, in space October 4, 1957.
Americans were somewhat alarmed and thought the Soviet Union must have ballistic missile capabilities that caused a threat. Politically, and perhaps militarily it put the Soviet Union ahead of the United States as a superpower.
On November 3, 1957, the Soviet Union launched Sputnik II with a dog, Laika on board. The United States congress immediately funded the Explorer project with Wernher von Braun at the helm. The space exploration race was on!
von Braun, the top German scientist entered the United States in 1945 right after WWII ended. Remaining in Germany would have cost him his life and would have cost the United States dearly in the space race.
By 1946, von Braun had already begun assembling and testing V2 rockets, many of which were manufactured from parts brought from Germany after the war ended. By 1952, he had launced 46 rockets. The principles used of the V2 rockets were used the rockets through the Apollo space program.
On January 31, 1958, the United States inserted its first satellite into orbit. It has a scientific payload and was a light and useful spacecraft. With the Explorer, a magnetic radiation field around the earth was discovered.
In July of 1958, the National Aeronautics and Space Administration was created. We know it as NASA. It was a successor to NACA, or the Space Act as it was commonly called.
Rocket development continued at a fast pace and the first man to travel in sub orbital space was Alan Shephard in May of 1961. Yuri Gagarin had already achieved the honor of being the first man in space in April of 1961. John Glenn orbited the earth in February of 1962 in the Friendship 7, from the Mercury space program.
The rocket that launched Shephard into space was the Redstone. It was the grandfather of the Mercury project launch rockets.
At the same time Redstone was used in the Mercury project, the moon rocket, Saturn was in development stages. It had its first lift off in October of 1961. Eleven unmanned spacecraft were released; research and development took place before the Saturn was ready for use in the Apollo space program.
Apollo was a great success. From there, we saw the development of the space shuttle, the space station and the rumors of travel to Mars.