Your adventure begins in Ered Luin, in an area known as Thorin's Gate. Talk to Gloin and go where he told you to (it's marked on your mini-map by a golden ring), then enter the mine. Talk to Gimli here. Follow the tunnel until you meet Otur, who is fighting with a small creature. Speak with him and take a weapon from a satchel nearby. Equip it, and then talk to the dwarf once more. Follow him. You'll see a scene in which one of the miners unleashes a troll. Talk to Otur and he'll open a gate for you. Go through it. You'll now face your first enemies - Cave-Claws.
Go through the tunnel, beating 5 creatures total. You'll see Gimli fighting with a troll - when you approach him, Gandalf will appear and save the day. Talk to the wizard to finish this instance (instance is a story-driven quest in which you are outside the basic game world and don't meet other players).