The Cold War was not a fighting war. After WWII many US politicians tried to make Americans believe that the USSR had planned to take over the entire world. There was an 'arms race' between the US and USSR, and both countries practiced imperialism, taking over smaller countries, meddling in their politics, picking their political leaders for them. Both countries saw the other as a threat and used the threat to justify their own imperialism. In the US we used the threat as a justification to spend more and more on arms and weapons, in fact from the time of WWII we were on a constant war footing, spending more on weapons than any nation in history that wasn't at war at the time.
During the Cold War we fought many small wars, starting with the Korean War. Sometimes these were called 'proxy wars' because we were supporting one side and the USSR supported the other, so it was a chance to compare how good our weapons were to the Soviet's without actually having a big war with them. Vietnam was one of these, but there were many others, in Central America, Africa, etc.
The Vietnam war grew out of control. It was a 'quagmire', something we hadn't experienced up to that point, a war we couldn't win and couldn't quit. President Nixon ran for president in 1968 mostly based on his promise to end the war. He said he had a plan, but he couldn't divulge the details before he was elected, for fear of tipping his hand to the enemy. It was called 'Nixon's Secret Plan'. But in fact, once he was elected it was clear that his plan was simply to declare victory and pull out.
For his own political reasons, Nixon kept the Vietnam War going for another four years, finally announcing the peace agreement just before his re-election. Nixon's own chief of staff, H.R. Haldeman, said in his memoirs that the ONLY reason Nixon kept the war going was that ending it would have caused a recession and threatened Nixon's re-election in 1972.
As for the Cold War, Republicans say it ended only when the USSR failed, which happened under the presidency of GHW Bush, the president after Reagan. But the Republicans wanted to give Reagan credit for it by saying that Reagan's huge increases in defense spending had caused the Soviets to go bankrupt trying to keep up. In fact this is not true. There were not corresponding large increases on Soviet military spending, and what increases there were had been planned years earlier. The Soviet Union collapsed under its own weight--something that's now happening to us at home.