Accepting and completing tasks, or quest, issued by clients allows you to get rewards. Some people may ask you themselves to undertake quests, but you can also check for them on the various towns’ bulletin boards. Some quests offer particularly enticing rewards, so try to complete as many as you can.
Each town's’ bulletin board will have a list of different entries for quests that include information pertinent to them, such as the objective, the name of client, and the reward. You can undertake multiple quests at once, and none of them a date by which they must be completed.
Should you ever forget a Quest's objective, you need look no further than the quests sub-menu under the Resource section of the main menu. Once you complete an objective, you, must report back to either the client who requested your assistance or the town bulletin board. Otherwise, you will not receive any reward.
Most quests in the game will require you to collect rare drops from normal enemies you can encounter in the field. There are several specialties in the game that can increase your chances of getting drops and loot from defeated foes like Zoology, Entomology, Anthropology, Ornithology, etc. However, the best specialty for farming is no other than the Ocarina specialty that you can learn by getting the corresponding book from the side quest "Collecting the Metal that Plays Second Fiddle". (Requires x5 silver)
This very useful specialty will respawn all nearby enemies in the immediate area where you used it. This will save you a lot of time and effort since you don't need to run around to look for specific enemies or wait for them to respawn. Activating the Ocarina from your Specialties menu will not consume anything so you can use it as much as you want. This specialty is also good for power-leveling since you can practically stay in one spot to farm certain enemy groups/types.