Taken from Wowpedia:
The Human and Dwarf Paladins today are really the same, they just lack the 'Order of the Silver Hand' tag."The Holy Order of Northshire Clerics, led by Archbishop Alonsus Faol, was devastated during the First War. Seeking to create an order that would suffer less attrition in battle, Faol recruited noble knights (to train in the ways of the Light) and members of the priesthood (to train in martial combat). In Alonsus Chapel in Stratholme, Uther the Lightbringer was appointed the first of the paladins. He was followed by Saidan Dathrohan, Tirion Fordring and Turalyon (candidates recommended by Alonsus Faol himself) as well as Gavinrad the Dire (a candidate proposed by Lord Anduin Lothar). The Order of the Silver Hand was thus born."
Blood Knights, Sunwalkers, and Vindicators function a little bit differently, but to the same effect.
Whereas the first order of Death Knights were made from the souls of Shadow Council Orc Warlocks placed the corpses of Stormwind Knights.
The Death Knights of the Scourge came originally from jaded and bitter Paladins of the Silver Hand, later they would also be reanimated Warriors and Paladins.
So basically, all it takes for a Priest to become a Paladin is a little bit of Sword training. Warlocks and Demon Hunters, while they share some visual similarities, are completely different (A Warlock can't just pick up glaives and become a Demon Hunter.)