Dragon's Watch has 14 Regions, each of which contains Normal, Hard and Nightmare level quests. At the time of the game's launch, the first 7 regions are live.
The last Normal level of each Region is a Boss quest. Beating the boss will allow you to rescue the dragon egg. Doing that unlocks the next Region, and also unlocks the Hard quests for the completed region.
The 'Next Quest' button takes you to the next quest in a linear sequence. If you can't complete a quest, you should go back to the Heroes screen and fuse Heroes to increase their strength, or to the Summon screen to summon more Heroes. You can also replay completed quests, to get more loot - new Heroes and Ingredients. To choose a specific quest, click on the appropriate Region icon on the world map, choose Normal, Hard or Nightmare quests to get to the Quest Select screen, then select the quest you want to play. This will take you to the Team Select screen, ready to choose your squad for battle.
When you have completed a Region, you can assign a Watch to it. This allows you to collect rewards from that Region even when you're not questing yourself - your Watch Heroes will do it for you.
The Vortex, shown at the bottom of the World Map, contains different quests, with different rewards, for each day of the week.