There are a variety of Quests in game. Usually an NPC with question mark means you can accept a Quest from it and an NPC with question mark means you can turn in your completed Quest to it.

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Tap on the exclamation/question mark to accept/complete the Quest. Generally, you can only have one Main Quest and one Sub Quest at the same time so you don't have to worry about missing anything. The normal quests will not come back after completing, but loop quests will be repeated (as the name suggests).
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All Quests are displayed in here separately (Main or Sub) in the form of: Quest Description, Quest Objective and Quest Reward.

Achievements range from enhancing gear, taking part in events or leveling up skills. The rewards usually correspond appropriately to the achievement. For example, upgrading magic card star gives you magic card fragments as a reward.

Completing Main and Sub Quests are one of the fastest way to level your characters.