The US did two major things to react to this. One was to try for a way to ensure nuclear peace. The other was to try to be stronger than the Russians.
The first thing that the US did was to propose a plan where both the US and the Soviet Union would turn over all their weapons to the United Nations so that neither nation would be able to use the weapons.
But at the same time (especially since that proposal went nowehere) the US started working on bigger bombs -- hydrogen bombs.
The US also adopted the ideas of NSC-68, which called for larger conventional forces so as to have an option other than nuclear war.
Realizing that the nuclear monopoly was over, and that this could quickly spiral into an expensive and dangerous arms race, the US reacted to the news of a Soviet bomb by putting together a plan to offer to turn over all weapons to the UN. This offer was rejected by the USSR, and an arms race ensued. Our secondary reaction, then was to develop our technology further, and a hydrogen thermonuclear device along with nuclear missiles would become realities within 6 years of the Soviet bomb test.
The Cold War had already begun, but this accelerated it into a new and much more dangerous phase.
The previous two answers correctly identify the two central responses of the United States to the fact that the Soviet Union did indeed detonate a nuclear weapon of their own. The United States at first tried to convince the Soviet Union to hand over their nuclear weapons to the United Nations. The US would agree to do the same. The Soviet Union did not agree to the plan, and the United States couldn’t have been too surprised. The plan that suggested that the US would turn in its nuclear arsenal to the United Nations had been in existence since 1946. It was called the Baruch Plan. The United States agreed to the plan on the condition that all other countries pledged to not produce nuclear weapons and agreed to a system of inspection as well. The Soviet Union rejected the plan then, so they weren’t likely to accept a similar plan after developing their own nuclear arsenal.
The main response of the United States went along the lines of quantity and quality. The US began to increase funding for developing the size of their nuclear arsenal and the quality of the arsenal. Fission bombs were adequate, but the United States began putting much more focus on developing thermonuclear weapons. Amazingly, the first test of the hydrogen fusion bomb was carried out in 1952. A year later the Soviet Union tested their first hydrogen bomb. By 1980, seven nations had thermonuclear weapons.