It is not at all clear what our “mission of cooperation and support” is. Therefore, I will try to answer this question in a generic way. Please feel free to clarify if you wish.
In Vietnam, the US was trying to cooperate with and support the government of South Vietnam. In recent years, we have tried to do this with the governments of Iraq and Afghanistan. Let us think about the military implications of Vietnam for such missions.
One clear military implication of Vietnam is that we need to do a better job of training and building the indigenous armed forces. The ARVN in Vietnam were simply not good enough to be a major source of support for the US. It is important to have significant support from local troops.
Perhaps the most important military lesson from Vietnam, however, is the need to move away from the sort of “body count” goal that American troops had in that war. This was simply not an effective way to fight. In that war, we demoralized our troops and our public by fighting hard for pieces of territory only to give them up because all we really cared about was “body count.” This shows that it is important to try to win and hold actual territory. This is good for morale for American soldiers, American civilians, and the local populace.
Thus, we can see at least two major military lessons that we can learn from Vietnam and apply to later missions.