Talk to Postman Took. You'll get your first quest. Accept it and leave the post office using the door behind your back. Go along the path until you see Bounder Boffin. Talk to him and examine the nearby Satchel. Take a weapon from it and equip it - press (I) to open the inventory screen, then double-click on the desired weapon's icon. Talk to Boffin again and wait for him to open a door before you.
Clear the way killing both spiders (you attack an enemy by right-clicking it), then talk to Boffin again. Follow him until you see a huge web. When Boffin starts to cut it, you'll have to deal with more spiders. Follow your companion when you're done. He won't be able to open the house's door, so talk to him again and go even further. You'll see a short scene in which you'll be saved by Amdir and his men. Wait for the NPCs to finish talking with Boffin, then talk to your saviors. This way you've completed your first instance in the game (instance is a story-driven quest in which you are outside the basic game world and don't meet other players).