If there is an acceptable quest, you can find it in the "Quest Tracing" and the font color will be green. You can also find an "!" on the mini map and on the head of the NPC.
Click on the NPC which has a quest and you can find the details of the quest. Click on "Accept Quest" to accept the quest.
When you complete a quest, the font color will become green again and the "!" on the head of the NPC will become "?". Click "Complete Quest" to complete the quest.