Varrock East Mine - Teleportation Menu > Cities > Varrock (Page 1) > Varrock East. Run through the east gate and then south to the mine. The guards above the mine have to be paid 25,000gp to pass.
This method costs 25,000 gp, taken from vault/inventory.
Alternatively, teleport to varrock via the same menu, run east to the fountain, south until you reach the walls of the city, then follow them east until you hit the mine. Or use the modern spellbook to teleport to the fountain and follow the same path.