Dwalin has one last intro-based task for you. Before accepting it, make sure you've completed all previous quests, as you won't be able to go back once you do it. Also, pay a visit to a trainer and learn all available skills (trainers are in The Berghold and in elf encampment). If you've done everything correctly, your character should now be level 5 or 6.
When you're inside the dungeon, talk to Dwalin. Your task is to make sure he's safe. Follow him and fight off the Goblins that attack you from time to time. When you get separated from the enemies by the flames, turn around and right-click on Tub of Fetid Water, then approach the fire and your character will automatically put it out.
A while later you'll be facing Gormr Dourhand and his fellow dwarves. When you defeat them, follow Dwalin even further. At the end of the dungeon you'll face Marecc and a Goblin Leader. Try not to get attacked by both enemies at once - let them attack Dwalin and then get attention of only one of them. After the battle talk to Elrohir and Dwalin. You'll end up outside the stronghold. Your reward for this quest is an item called Map Home. Once every hour you can use it to teleport to a town that's currently marked as your home location.