Not nearly enough.
We saw how easily Germany was lead down the primrose path to total war, and we still listen to ideologues and demogogues. Every democracy has its loons, cynical bastards, chickenhawks, and right bastards. It wouldn't be the same without them. But when Rush Limbaugh can define a whole genre, what's gone wrong?
We saw how brutally the majority can treat the minority, and there is still debate over whether gay marriage is a good idea. This isn't at all comparable in one way: Nobody worth listening to is advocating the massacre of gays. But in another way, it's a simple continuation of the same old crap minorities have had to put up with forever. We should have buried prejudice and bigotry alongside the memorials in remembrance of the death camps, but, of course, we didn't. We cry out of one eye while we glare with the other.
We saw Dresden, Tokyo, London, and Berlin, and we still call for total war and bombing them back to the Stone Age. This truly is an aspect of things that hasn't changed: People who have the public eye seriously think that turning more cities into rubble and more civilians into paste would solve our problems. There are people who would firebomb Falluja, Najaf, Baghdad, and Tikrit simply to wipe them off the map and teach them a lesson. Which means we haven't learned our lessons.
We saw how bad Stalin was, and we still say "He may be a son-of-a-bitch, but he is our son-of-a-bitch." when creating foreign policy relations with madmen and mindlessly brutal gangsters. Do we need to actively back these monsters to maintain peace and order in the region? Some would say so, but think of this: Our two most recent wars have been against the people we installed and gave material support to.
We saw how easily Hitler snapped up Poland, Austria, and The Sudatenland when he was unchecked, but we still let Realpolitik trump common sense. How many powder kegs are we ignoring right now? Which minor local madman will become the next Adolf Hitler or Che Guevara? How many of these beastly people are we responsible for?
We learned jack shit. As usual.
Maybe we aren't as enthusiastic about nuclear weapons. Maybe we don't automatically trust every leader. But that's about it.