The easiest way to complete this quest is to talk to Farmer Fred and buy 20 balls of wool at a general store or Grand Exchange for 220 coins, then just give them to him and the quest is complete. Note that he will not accept noted wool, but only the actual item(s). This is not advised for players who enjoy the satisfaction of a quest.
If you feel like actually doing the work, follow these instructions:
First, grab the shears off the table in Farmer Fred’s house.
Next, talk to him. Tell him that you’re looking for a quest. He will give you one, and you will need to get him 20 balls of wool.
Leave his house and enter the fenced area on the east side of his house, which is the sheep pen. You can either use your shears with the sheep or left-click on them to shear them. Sheep with wool will look puffy, and sheep without wool will look skinny. If a sheep has no wool, wait a few minutes and the wool will eventually appear on them. Sometimes they will run away from you if you try to shear them; just keep trying and you’ll eventually get their wool.
Once you have 20 pieces of wool, you must now spin them into balls of wool. Enter Lumbridge Castle and head to the second floor. In one of the rooms is a spinning wheel. Click on it and spin the 20 pieces of wool you have. Note that you will receive 2.5 crafting experience per wool you spin.
When they are all balls of wool, go back to Fred the Farmer to finish the quest. If you do not have the space to carry all 20 pieces of wool, you can make smaller deliveries until you reach the required total.
Reward: 150 crafting experience, 60 coins