Quests are another integral part of Sword Quest.(2nd part of the title really..)
There are 6 main quests, 1 minor part of the game, that while not a quest does require you to complete requirements and you do get rewards.
The Hero Quests are the main storyline quests. This gives you a reason and some motivation to improve your skills as it allows you to see the direct impact you have on another because of your rising skill. There is no time limit on your Hero quests.
The Kingdom Quests as it states are randomly picked. It can be any of the coin bought swords from the shop(up to what you have unlocked), and will reset at the start of every new day if you have completed the previous one. There is no time limit on this type of quest.
The Candy(Weekly) Quests are similar to the kingdom quests in that it picks a random material. The biggest difference though is that the weekly quests do have a time limit. You have 1 week. The other difference is that not only does it only require 2 swords from the material it asks for, but it also gives you 1-2 free materials.
The Special Quests are the challenging quests that you do for Feludias more obscure citizens. These quests can be unlocked by completing X number of total swords, or by turning in a high level sword from a specific material. The special quests generally give either gold, hammers, or materials as the reward, and often a mix of the 3. The clues for solving these quests can be found in the Mangas. There is no time limit on your Special quests.
The latest Adventurer Quest (Guild) was released on May 9th, 2013. These quests can be unlocked by completing 3 Stars on any of the participating locations with 3 Boss. Your goal will be to vanquish these boss and as a mean to motivate you to advance the adventure mode for better rewards. The added bonus is all of these daily quest gives over 1000 Exp to both you and your pets that'll help players who have no more first kill bonus to level up.
One quest that is often over-looked, but is part of the game is Saving Swords. The pink-haired lady who helped you out at the beginning initially challenges, and then encourages you to create more swords. As you do so, she gives you rewards for reaching certain milestones.
Goals were added with the 7/3/13 update and help provide the player with a bit of direction and makes it easier to advance. Regardless of your level, all players have the same goals.
Heroine Quests were added with the 8/27/14 update. This unlocks Heroic Route(which itself hasn't yet been unlocked) and takes the player on an adventure. This was introduced for the newer players and helps explain why exactly you decided to leave your workshop of just creating swords to exploring the world.
Last are.. Seals. These are different things you have to complete to get rewards. The objectives you have to complete can range from the number of epic swords you create, to the number of workshops you own, to even the number of times you have failed a fusion. There are 4 parts to each objective. The reward is the same for each type. Copper, Silver, Gold, & Divine. You want to get to Divine Status for each objective(19) as that rewards you with 1 Divine Metal.