Rune Mysteries is mostly important for unlocking the ability to mine essence for the Runecrafting skill. With proper teleportation methods, this quest is very easy and short, but for many new non-members without teleports, this quest can be very "annoying", seeing that you run back and forth a lot.
To start this quest, talk to Duke Horacio, who is on the first floor of Lumbridge Castle, and ask him for a quest. He will give you a strange talisman, and asks you to take it to the head wizard of the Wizards' Tower for identification. Exit the castle, and go to the Wizards' Tower, which is south-west of the Castle and south of Draynor Village, and go meet the head wizard.
Look for the wizard Sedridor in the basement and then talk to him. He will confirm to be the head wizard and take the talisman given by Duke Horacio before giving you a research package, which you have to deliver to rune expert Aubury in Varrock. Aubury lives south of Varrock's east bank where he has his own Rune Shop.
Getting There: From the Wizards' Tower, you can use the Varrock Teleport to greatly shorten the trip, as it is a long walk from the Tower to Varrock. Alternatively, you can use the Lumbridge Teleport, then use a canoe to Varrock, (if you have a level 12 woodcutting) or the Varrock Lodestone.
Go to Aubury's Rune Shop and talk to Aubury. He'll want you to deliver some notes back to Wizard Sedridor and also ask you if you would like to be teleported directly to the Wizard's Tower. This teleport will bring you outside the Wizard's Tower. (He sometimes will not ask)
It's possible to train Runecrafting without actually completing this quest, but you can't mine Rune Essence. This is easier because of the introduction of the Grand Exchange, because you can trade rune essence or pure essence without having to have completed the quest. Methods of training include Runespan, Runecrafting during Dungeoneering or buying essence and then runecrafting at altars.
After completing the quest, if you talk to Juna, your character will say "...So, I brought Aubury's notes to Sedridor the Head Wizard and, from then on, I was able to mine rune essence."
Before an update, players used the drop trick to gain multiple Air talisman to make considerable amounts of money. It's now a strange talisman.