Why did the United States and the Soviet Union become involved in an arms race?

The US/USSR arms race was seeded in the use of Nuclear weapons against each other

Explanation:

The tension between the US and the USSR was something that had been building for years throughout WWI-Vietnam (even longer till 1991 and the dismantel of the Soviet Union). The use of the atomic bomb by the US in WWII    even after the alliance the two countries had, started the competition. There was an amount of distrust since the US didn't disclose their plans to drop the bomb on Japan, to the USSR.

After the end of WWII, the two powers were on opposite ends of their views on communisim. The USSR sought to expand their rule over Europe and the US was buliding their defences against that possiblity. Knowing the US had nuclear weapons, the Soivets tested their Atomic weapons in 1949, which was the true beginning of the Cold War. The US went on to test the Hydrogen Bomb, in 1952 and the USSR followed suit in 53. The arms race turned into the famous Space Race, which the USSR led by launching a satellite into space, pushing the US to put a man on the moon by the end of the 60s.

Major events:

The Bay of Pigs, 1961 which was an unsuccessful invaision of Cuban exiles backed by the US leading to tension between the US and Castro

The Cuban Missile Crisis in 1962, between John F. Kennedy, Fidel Castro and Nikita Khrushchev.