In response to the Russians' more advanced space program, Eisenhower signed the bill to form and fund the United States' own space agency, the "National Aeronautics and Space Administration", or "NASA". The purpose of NASA was to give engineers, scientists, and pilots (future U.S. astronauts) the resources and support they needed to compete with Russia in what would come to be known as, "The Space Race".
Advances in Space Exploration during the "Cold War" period were considered very important to both Russians and Americans. This was because the rocketry and guidance systems each country developed could also be used as horrific weapons against its enemy. During Eisenhower's two terms in office between 1953-1961, both Russia and the U.S. made great progress toward launching humans into space, generally with the Russians in the lead, though neither country had yet made a moon landing. Landing a man on the moon was, at that time, considered to be the finish line of "The Space Race" that both countries were trying to reach. Both countries were hoping they could boast that they were the first. (See this Space Exploration Timeline:
https://bit.ly/3kdcS5V)
"The Cold War" period was a time when both Russia and the U.S. continued to develop more and more nuclear missiles, pointed toward its rival country's most populated cities. Additionally, both Russia and the U.S. tried to establish their influence (and create alliances) with other countries that were strategically important if another world war erupted, which was considered likely then. Throughout his time in office, Eisenhower worked tirelessly to try to ease tensions between the U.S. and Russia. Each U.S. President since Eisenhower has tried to do the same.
On August 8, 1953, the Soviet Union (Russia) announced it had hydrogen bomb capabilities. In August 1954, President Eisenhower signed the Atomic Energy Act to authorize the international exchange of information on the peaceful uses of atomic energy, and endorsed the development of commercial nuclear power.