Unlock quest by leveling the Inn in your city with investments! This will also unlock additional levels of each quest. If you are unable to send heroes to fight a boss, this may be the cause. Upgrade the Inn to the next "New Rewards" mark for that dungeon to continue.
Click on a quest slot on the bottom of the Inn page. Select the quest you would like to send heroes. A screen will pop up with the Group Power Requirement. Heroes have an individual recommended power for each quest. This is denoted by a colored emoticon above the hero’s head.