Pay a visit to the NPC Angus in Port hope and join the society, he will then ask you to bring him a dwarven pickaxe (easiest way is to buy it from the market).
If you don't want to buy the pickaxe from the market or it is currently not available this is the easiest way to obtain it. You don't have to do this if you already have a dwarven pickaxe:
"You can ask NPC Uzgod in Kazordoon for a dwarven pickaxe, to which he will reply that he will exchange one for his beloved Family Brooch (you must talk to him before you can get it). Go to Dwacatra, near the main gate, and search for a hole in the center of the prison. Down there are 3 Dwarf Soldiers and 1 Dwarf Guard. Kill them and find the Family Brooch in the sealed door on the south. Go back to NPC Uzgod and turn in the brooch to receive the pickaxe.
If you choose to go to Dwacatra you will need a Shovel and Key 3802."
Turn in the pickaxe to NPC Angus and you are now a member with the Novice rank of the explorer society.