Quests are tasks players can complete to earn rewards such as coins, rebirth tokens, crates, eggs, and special limited hats. As you complete more quests, you will be given better rewards.
In order to start a quest, you must meet with a Quest Giver; an NPC that will give you a quest to start. Each world has a Quest Giver. For completing a Quest Giver's quest line, you will be given a special limited hat.After completing a Quest Giver's quest line, you can still be given quests by the Quest Giver, but these quests are classified as "Unlimited Quests," where you can still earn rewards, but no special prize such as the special limited hats.
If you want to check your current quests, click the "Quests" tab on the bottom right of your screen.
Please note that you'll have to collect more ores than the quests requires because the quest has a chance of not updating even if you mine the required ore.