There are many lessons to be learned from this war.
For example, from the early parts of the war, one lesson that should have been learned was to avoid tying the country diplomatically to a weak and unpopular regime. The US put itself in a position where it had no good way to disengage from the Diem regime, whose policies made it extremely unpopular with its people.
At the end of the war, a lesson learned was that it is always better to bargain from a position of strength. The North Vietnamese were essentially forced into a peace agreement (though it was certainly one that worked well for them) because the US was in a relatively strong position. It had hurt North Vietnam militarily and it was improving its relations with North Vietnam's communist allies, China and the USSR. Because of these US strengths, the North Vietnamese agreed to peace terms that they might otherwise have rejected.