Different Americans feel that they have learned different lessons from this war. This is because different Americans have different attitudes about why the war was lost.
One thing that just about everyone agrees on is that the US needs to send troops into situations only when those situations have clear ways to know when we have won so we can leave. This was a major problem in Vietnam and it seemed that we had learned that during the first Gulf War. However, we seem to have ignored that lesson in Afghanistan and Iraq.
Just about everyone also believes that we learned that we need to support our soldiers regardless of what we think of the war in which they are fighting. Soldiers returning from Vietnam were treated badly. This lesson has clearly been learned and applied.
More controversially, some think that we learned that wars can be undermined by public dissent. This is why some said that people who criticized the invasion of Iraq, for example, were giving aid and comfort to the enemy. They feel that we lost Vietnam because our public was not patriotic enough and did not support the war firmly enough.
Others feel that we learned that we should not try to impose our ways on other people. They feel that Vietnam showed that we could not make others like our system of government. Therefore, we should only get into wars that are not aimed at “nation-building.” Again, we seem to have forgotten that lesson in Iraq and Afghanistan.
Thus, there are multiple possible lessons that you can say we learned from this war.