To start your task, you need to go to NPC Grizzly Adams, located in the busiest cities. In the attached image to the side, we have NPC Grizzly Adams located in Thais near the Depot. To start your task, type hi, task, dragons, yes. You will start the task of a specific monster and will have to kill a certain number of this monster to complete your task. Remember that you can choose up to 5 tasks to do simultaneously. If you want to see the available task options, type hi, task, task names.
After finishing your task, return to NPC Grizzly Adams and type hi, report, dragon, yes. If you killed the required number of monsters, you will receive your reward.
(Click on the image to enlarge.)

BOSS ROOM: In certain tasks, when you finish the report, you will gain access to kill a specific BOSS and will be teleported to the BOSS ROOM automatically. When you step on one of these orange tiles, you will enter an exclusive room, and the respective BOSS will be summoned. If you leave the room before killing the BOSS or if you die, just talk to the NPC and type hi, boss room, yes.
(Click on the image to enlarge.)