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1. As the heading says, your first mission, if you choose to accept it, is to find the Fighter's Guild on the west side of the city, and then talk to Hasphat Antabolis, located downstairs, about the Nerevarine and Sixth House cults.
2. Hasphat will ask you to do something for him in return for the information (which is the case in the majority of like-quests in the game) - find a "Dwemer Puzzle Box" inside the Dwemer ruins named Arkngthand, a short walk from here.
3. After stocking up and doing whatever else you feel like doing (there is an almost overwhelming amount of things you can do to pass the time besides the Main Quest... I suggest buying a healing potion or two), head south out of the city gates, and then east. Follow the path until the first fork, and take the left branch. You'll see a signpost on the right, so head to the up the path on the hill to your right toward "Molag Mar."
4. Your first major battle, if you choose to do this quest first, is likely to be coming up. There's a bridge ahead. At the end sits a conjurer with some decent equipment. When you're in range, he'll begin moving toward you and summon a skeleton, so be prepared. Even at level 1, he shouldn't be too tough, especially if you brought a healing potion or two. Focus on him rather than the skeleton if you're having difficulty.
5. This is a general tip. Search the crates at the end of the bridge for a chance at some valuable loot, and remember to search every crate, box, chest, container you come across, because you never know what you'll find.
6. Past the bridge you'll probably fight one or two Cliff Racers that will fly at you from above. Again this encounter should not be too difficult, but don't despair if you die... the game WILL get easier, probably much easier.
7. On a path to your right after the bridge, you'll find a crank on the left side. Turn the crank (activate it) which will open the doors to Arkngthand just a little further up the hill. Go on in.