The sewers of Ardougne are located west of the Ardougne Marketplace in East Ardougne. To enter you must open then climb down a man-hole.
You will find yourself in an area with a few rats and then a locked gate. You must 'picklock' this gate (Level 31 Thieving) in order to access the iron and coal mine further down in the sewers. This mine is guarded by a few zombies so if you are a lower leveled player, be aware.