Honestly, not a lot.

One of the crucial turning points for the US role in Vietnam came under JFK's watch, specifically the South Vietnamese military coup against its democratically-elected President Diem. Washington was aware of the coup planning and offered tacit support to its plotters, because Diem and his brother were ardently Catholic and pursuing pro-Catholic policies in Buddhist-majority Vietnam. (This religious tension inspired several Buddhist monks to self-immolate in protest; one of the most famous ended up on a Rage Against the Machine album cover).

When Diem was removed, all hell broke loose in South Vietnam's government. The first coup participants were replaced in weeks by a second coup (also from within the military), which was in turn overthrown by a third a couple of years later. The citizenry, already distrustful of the pro-Catholic Diem government, did not trust the subsequent juntas at all.

In short, Diem's removal created a power vacuum in South Vietnam at the exact time a strong, functional government was most needed. That vacuum would come to be filled increasingly by the US, creating a snowball effect in which the South Vietnamese would be unable or unwilling to perform a given task, so the US would simply step in to fill the gap. Increased US presence, in turn, further undermined whatever legitimacy the South Vietnamese military juntas had, so they'd become less effective, warranting further US involvement in the region.

Given this power vacuum, the US was given two basic choices: leave Vietnam entirely, knowing that the ineffectual and unpopular military government would rapidly collapse (which is what happened when the US actually pulled out), or stay and fight the Communists. The latter option proved attractive due to Domino Theory and the general momentum described above.

All of that started under JFK. The first coup plotters asked for and received Washington's support for their plans in August 1963. The logic behind continued involvement began under JFK's watch and continued after it. There was no realistic path for the US to exit South Vietnam after Diem's removal, because there were no adequate replacements and the US was involved behind the scenes, so it couldn't simply burn the metaphorical house down and just walk away. JFK started a ball rolling that his own advisors warned him he'd be unable to stop.

Sources:

a bunch of diplomatic cables which are now public record. You, too, can read many of the raw cables sent from Washington to its South Vietnamese ambassadors along with internal decision-making stuff.

Pentagon Papers part IV.B.5 - A collection of Diem-specific memos

Foreign Relations of the United States, 1961-1963, Volume III, Vietnam, January-August 1963

Foreign Relations of the United States, 1961-1963, Volume IV, Vietnam, August-December 1963

And all the sub-memos listed / referenced by the above.