The Vietnam War did not begin with one incident like the German invasion of Poland in 1939 or the Japanese attack on Pearl Harbor. Instead, the Vietnam War arose gradually out of Vietnamese nationalism and their attempt to resist being colonized.
What we call the Vietnam War first started to arise during WWII. Many Vietnamese had been very unhappy about being colonized by France. When Japan took Vietnam from France, they saw this as an opportunity to move toward independence. At that point, a group called the Vietminh (led by Ho Chi Minh) opposed both the French and the Japanese. When the Japanese were defeated and left Vietnam, they declared independence and resisted French attempts to recolonize the country.
We can say that the Vietnam War began at that point. The French, with American support, fought the Vietminh insurgency. This war is sometimes called the First Indochina War. The French eventually withdrew from Vietnam in 1954. At that point, the US stepped in. Vietnam was divided into two countries. North Vietnam was controlled by the Vietminh, who were communist as well as nationalist. The US supported a non-communist regime in the South.
This is how the Vietnam War started for the US. The North helped communist rebels in the South (known as the Viet Cong) fight against the US-backed government. The US provided more and more support for the South Vietnamese government until eventually the US became fully embroiled in what we call the Vietnam War.