Syngman Rhee(south korean dictator) absolutely did want to take all of korea. he openly voiced this and provoked NK where ever possible.
due to his provocative actions, the us actually did not allow a bunch of heavy equipment to be given to SK, things like tanks and artillery, as the us did not want a war there that could involve the soviets so soon after ww2.
the north under Kim, of course also wanted to take all of korea, and they were no less peaceful, clashes were frequent along their borders. however the communist enjoyed high support across all of korea and thus did not need to resort to massacres the way that the south did and the soviets were all too happy to send kim all the tanks and industry he needs.
thats said. its the north that initiated the korean war. while the north is painted as a aggressive, expansionist power, i would disagree since this was a national reunification war and korea's division was because superpowers drew a line in the middle of their country not because they voted to split. and while SK obviously turned out richer and more prosperous, at the time, the south was a murderous regime that killed hundreds of thousands of koreans to simply silence them because communism had wide support.