Did most of the people of South Vietnam support the Viet Cong and NVA, or was there some loyalty to the South Vietnamese government?

There was a great deal of support (even if it was qualified in wanting the government to improve) for the Republic of Vietnam. For this seems quite obvious. But I certainly understand how one can get this impression, as it is quite easy to encounter simplistic portrayals of Vietnam that were produced by the intense politicization of Vietnamese history during the anti-war movement. Keep in mind that during both the First (1945-55) and Second (1956-1975) Indochina Wars, the primary combatants were Vietnamese -- on both sides. What I mean is that the non-communist Army of the State of Vietnam and its successor after the 1954 Geneva Accords divided Vietnam, the Army of the Republic of Vietnam (ARVN), were doing the much of the fighting and dying. There was a vibrant society in South Vietnam and many people had misgivings about the government, that isn't the same as saying they supported the North or insurgents. In the late 1960s there was even nostalgia for the Ngo Dinh Diem period in South Vietnam. The demand of dissident South Vietnamese elites was more often for political reform and democracy -- something the other side would not deliver. The Viet Cong was not a rosy group. It used terrorism and coercion to accomplish its goals, as did their benefactors in North Vietnam, both during the 1945-55 and 1955-75 periods. The majority of people, particularly in the countryside, were caught in the middle and had to accommodate the circumstances they faced.

The other aspect I will note is that many intellectuals and youths sympathetic to the Viet Cong were very naive about the Democratic Republic of Vietnam. They did not think of their support for the Viet Cong/NLF as support for a unified state under an authoritarian communist regime. They believed they would have a say in a new government and help address problems in southern society (keep in mind there were a great deal of problems in North Vietnam at this time too, issues like press freedom, intellectual liberty, etc were tremendously worse there). Instead, after the fall of Saigon, the North Vietnamese authorities asked the Provisional Revolutionary Government/NLF to vote itself out of power. What began then was simply the extension of the communist party's control over all of Vietnam. The students and many other supporters realized that error and their relationship with the government quickly soured. That became apparent as southern Vietnam was plummeted into a police state and poverty in the late 1970s and 1980s, eventually forcing the party to implement the Doi Moi economic and cultural reforms after 1986.