What strictly military lessons were gained from the Vietnam War that would help you accomplish your mission of cooperation and support more effectively?
The nature of the question is more than strictly military. The idea of entering the conflict with cooperation moves into a political realm of action. When a nation cannot convince other nations of similar values in the need for military action, it might be reflective of more than simply military presence.
I do think that one military lesson that was effectively learned from the experience of the Vietnam War was that greater intelligence needed to be gathered before entering into full combat mode. There were so many unknowns present in the conflict that could have been understood clearly prior to military conflict. For example, the United States clearly entered the war with the belief that the conflict centered on the assertion of Communism on the part of the North Vietnamese. However, a clear military and historical understanding of the region would have revealed that they perceived the struggle as one of independence from Colonial interference. The North Vietnamese adopted the same military style action as they did against the French because they saw the battle with the Americans as another challenge to a Colonial power. Each escalation from Rolling Thunder to ground troops was a military action seen as seeking to establish Colonial control of Vietnam.This might also be reflected in the fact that American forces were largely alone in the conflict. Grasping the historical condition of the region might have altered military strategy and interaction.
Another military pearl of wisdom would have been understanding the geography and climate of the region. The intensity of weather conditions and topography took soldiers by surprise. Part of the challenges in equipping soldiers with so much gear in such a humid part of the world leads to quick fatigue and less discipline in the field. A better understanding that this region of the world was unknown for the young American recruits might have helped to alter military strategy. The idea that recruits could be trained "in the rear" in Vietnam, when increased deployments made "the rear" absent was part of this reality. The military understanding of the region, itself, and grasping the intrinsic "home field advantage" that the Vietnamese held in the conflict proved to be essential.
Finally, I think that the military configuration of victory needed to be better understood as a result of the conflict. The prevailing wisdom of the Johnson Administration was that sustained American military presence will result in victory. This was seen in Rolling Thunder, Westmoreland's demand for more troops, and the approach to the war after the Tet Offensive. The metric for victory was never singular in focus and clear in discourse. The perception that started the war was that it would be a traditional understanding of victory. This was radically changed when it was evident that the North Vietnamese had elements of its own to define victory. The attrition metric was far too nuanced to explain to the American public and simply made little sense given how the war's definition of victory was initially presented to the public. In the end, there needed to be a greater military vision in defining victory. This would have altered American military presence in the beginning of the conflict and would have also articulated a clear exit strategy, realities which were absent from the American escalation of conflict.