The Indochine war began after negotiations were suspended between the Viet-Minh and the French Republic in 1946 (the date is often the insurecction on the 19th december, but in fact, the French bombed Haiphong on the 23th november, some even trace it to the leave of General Leclerc, the military administrator, in 1946).
The US involvement only began in 1954 (even if Vietnamese soldiers fought along the US and South Koreans in Korea), because the US had "found out" a communist government in Vietnam wouldn't be very good. The US backed the South Vietnamese government to ignore the results independency negotiations of Geneva (1954). These treaties specified a general election had to be organized in 1955. However, given the Viet-Minh methods, this election was likely to result in a "crushing victory" for the Viet-Minh in the North, if any other party had been allowed to take part.
During the late 1950s, the southern government, which had never been elected (the French had put former emperor Bao Dai as Head of State short before the independency, but without organizing any election), became more and more unpopular and autoritary. At the same time, ex- and current Viet-Minh members formed the Viet-Cong.
1963 can be considered the beginning of the Viet-Nam war, as the year saw the first battle where US soldiers were involved, as well as the assassination of the South Vietnamese dictator Ngo Dinh Diem and the establishment of a military government without any reaction (im not sure about this) from the US government, who started a new aggressive policy towards North Vietnam (Kennedy wanted to reduce US involvement in Viet Nam, but Johnson changed this). And so it began.