Militarily, one of the major things that was learned in this war was that defense had the advantage in modern warfare. This lesson was applied very effectively in WWI, where trench warfare with modern weapons similar to those of the Boer War gave a huge advantage to the defense. Of course, the lesson was not learned well enough to prevent armies from trying frontal attacks on those defenses.
Politically, the Boer War taught that it is very difficult to defeat a popular native insurgency. This is particularly true when there is some amount of international sympathy for the insurgents. The British made a major mistake when they created the brutal concentration camps to hold Boers. This made them look very bad in the eyes of the rest of the world and helped make it difficult for them to win the war.