When Can I Do Quests?
When you are at Player Level 10. Quests are accessed from your Base screen, in the lower left corner. Quests let you battle for resources, recruits, gems, experience, ascension materials, battle items, or materials. They are time limited events. Quests use map energy. Beside the map energy for each level you can see a number to the right of a sword. This indicates a recommended team power level to try that level of the quest.
Just keep plugging away at these levels. There’s no rush, as there’s no prize waiting for you at the end. At some point you’ll hit a wall where it is too difficult to go further. Go back to previous levels to win materials you need to train up your team and your base to make you stronger. Depending on your playing style you can push forward with battle items (potions/weapons/tools, etc), but really, for the most part you will get there with patience and time. Around Province 19 or 20 it gets much more difficult to do levels without battle items.
Each stage on the map requires a different amount of map energy to do. Map energy is represented by the red flags at the top of your screens. It takes 10 minutes to regenerate a point of map energy. Each time you reach a new player level your max map energy goes up 1 point. (Edit! Previously believed that the map energy topped out at 28 points. Now know that you get another energy point at level 31.) You can refill your map energy from energy flasks that you can win from titans, mission chests, and mystic vision, or purchase from the store.
When you tap on a stage in a province you get a summary of what types/colours of opponents you will face, and a list of potential (not guaranteed items) that you can find here. You’ll see that some stages look like on the map. These stages are more challenging, and give out slightly better rewards.