During JFK's first and only term, America had provided military advisers to Vietnam (mainly at the urging of Air Marshall Ky). In 1963, Kennedy was considering withdrawing some/all of these US military advisers. He didn't get the chance.
As a result of JFK's assassination, Lyndon Johnson became President.
Just a few months later (1964) the Gulf of Tonkin event 'happened'. As a result of that (now known to be a false flag) event, Johnson declared the US to be at war with the North Vietnamese . That's when it got serious.
As to escalating the war in Vietnam, the biggest escalation was done by Nixon, who even authorised the carpet bombing of Cambodia.
