From Hello Kitty Online Wiki
Quests are a big part of Hello Kitty Online. If you're not socializing then chances are you're working on one of the various quests available to you. Most quests require you to fight monsters and gather various items from trees, bushes, deposits, etc. via your Farming, Gathering, Woodcutting, and Mining skills. Even more quests want you to create new items by combining one or more via Forging, Carpentry, Cooking, and Tailoring. A few quests even require you to beat a flash mini-game!
How to Start a Quest
Questing is simple. All you have to do is talk to a NPC! Be careful, not all NPCs have quests to give you. You'll know which NPC has a quest for you by looking above their head. If there is an exclamation mark (!), that NPC has a quest available. Talk to the NPC and a window will pop up giving a description, requirements, and rewards of the quest. Simply click accept (the checkmark) to start the quest. Accepting quests will place them in your Quest Log where you can view unfinished and finished quests, and remove quests.
How to Finish a Quest
When you've completed the requirements of the quest, a message should appear in the chat display saying, "Quest Name: Quest requirement complete". At this point, a checkmark will appear above the head of the NPC you must go to, to complete the quest. Please note that the starting NPC (the one who gives you the quest) and the final NPC (the one who gives you the rewards) are not always the same.
How to Drop a Quest
If you open your Quest Log, by either clicking the appropriate button on the side or pressing 'Q' on your keyboard you'll get a list of quests you've accepted and completed. The first page is the page of accepted--but not completed--quests. Find the quest you wish to drop and click the drop quest button at the bottom of the quest log.