How did the Vietnam War play a part in turning the united states in a more conservative political direction
Although the Vietnam and Korean Wars were extremely different in most aspects they both led to a turning away from progressive American values to an attempt to install the sort of narrowly controlled country that caused most American families to immigrate from Europe. The anti-war movement during the 1960s was right in the general idea that we had no real national interests in Vietnam, and that the "Domino Theory" was inapplicable. However, the movement was top-heavy with extremely left-wing groups and political coalitions with no grounding in reality. Theatrical displays such as the exorcism of the Pentagon were ridiculous, trivialized the real issues involved and offended people. It is important to understand, though, that Richard Nixon was not elected in 1968 because he was so conservative, but because he claimed to have a "secret plan" to end the war. Nixon was the peace candidate among the two major parties (at least after the assassination of Robert Kennedy).
It was the failure of Nixon to deliver that led to the later excesses of the peace movement, that and his administration's disregard for the laws they claimed to respect. Unfortunately, the reaction of the anti-war movement on the surface was so offensive that it frightened people into allowing the swing to the right, although at the same time the majority of Americans did come to believe that the war was wrong. The real problem was that the reaction of the right-wing and the administration came so close to actually fomenting a civil war here in America (combined with the Black Panthers, Yippie, Kent State, the "Silent Majority", etc.) that the war had to be ended. Still, the realization that our country had become so polarized instead of leading to more open public discourse led the administration farther to the right and the average voter allowed it, not knowing how else to deal with what really looked like looming anarchy. Much like the 1968 Tet offensive, it looked worse on TV than it actually was.
The normal economic drop at the end of a war naturally followed, and this plus Watergate led to massive disillusionment. Carter succeeded in restoring a certain amount of respect and confidence in the government, but the economy and the military had been ravaged by the Nixon and Ford years. Since Nixon's misuse of the CIA had caused loss of trust in them, also, that agency had to be restructured. Carter's administration succeeded at setting in place a large number of programs which did fix these things, but they took time to work. The fear Americans had about the world caused them to take shelter in the soothing rhetoric of Reagan, who of course took credit for all the programs Carter initiated (Strategic Defense Initiative, Unified Special Operations Command, restructuring the CIA, and a slew of economic programs). Carter's support for the Afghans fighting the USSR (and the eventual success of Truman's Containment Policy) led to the collapse of the Soviet Union, which Reagan also took credit for. All these things together have been the public face of a swing to the right in the decades since Vietnam. Unfortunately, below the surface is a much uglier face, the rise of the corporate state supported by government propaganda. Unlike Nazi Germany, however, we do not have a fascist government controlling the corporations, we have corporations controlling the government. That is an improvement.
Many of the perceptions and beliefs that emerged from the Vietnam War helped to foster a greater sense of Conservative political thought. Right or wrong, the perception was that the conflict belonged to President Johnson, and his willingness to commit troops in such a stark manner to the conflict was a result of his liberal belief system (In fairness, it should be noted that the expansion of the war to neighboring nations increased during the Nixon Administration.) The perception was that a liberal foreign policy that increased U.S. commitment abroad was a flawed one. This might help to explain why greater conservatism was sought in terms of foreign affairs, less advocacy to use force was brought out, in the period that followed the Vietnam Conflict.
During the Vietnam War, there was a lot of protest against the US participation in the war. A lot of the protest was very unruly. This really helped push the country in a more conservative way. That is because the protestors seemed to represent a really unAmerican point of view -- one that was completely disrespectful of the military and of authority.
Second, the war, combined with LBJ's War on Poverty, caused government spending to go way up. This helped lead to something of a backlash against government spending as seen in the Reagan years.