There are two different kinds of quests,normal and commissioned.

To obtain a normal quest find NPCs with scrolls on their heads.After select the quest you click on accept.To view your current quests you have accepted press "Q" to check your quest journal.

To obtain a commissioned quest you need to visit the bulletion board in Parka Shrine or Draco Desert Village.After you click it click on the "I want to receive a commissioned quest" and see which one you're going to obtain.To get a commissioned quest you have a pay a certain fee.It is different for each quest.Your reward for completing a commissioned quest is one reward out of five.For example if you obtain a commissioned quest the reward could say "Reward:90000,120000,150000,200000".Then when you finish you get one of those four rewards.