Objectives: Locate the Star Map
Korriban is a world dominated by the Sith; their main training academy is located here. Indeed, this very planet was where Exar Kun fell to the Dark side not many decades before the time of the game.
After you land, pester the mechanic on the docking platform for a bit of background information about Korriban and the Sith Academy. It would seem that the Star Map is likely located on excavations that the Sith are performing on the surface of the planet. Unfortunately, only Sith Academy members are allowed access to the excavations. Guess what you have to do next?
As you proceed into the facilities, you'll come across a couple of vignettes where you'll be forced to choose between violence and inaction. The first involves a group of acolytes attempting to gain access to the Sith Academy--you'll be asked to choose whether their instructor, one Shaardan, should let them live or die. The second involves a confrontation with a group of Sith apprentices. Despite appearances, you won't be able to fight them--yet. There aren't any alignment shifts as a result of this conversation, no matter what you choose.
Speak to the prospective Sith in the Czerka store to get yet more background--it would seem that there's a Sith Academy instructor that hangs around the nearby cantina that you're going to need to impress. Outside that same cantina, there's a Rodian named Lurze Kesh who offers you the Unfinished Business side quest. For now, head out the exit to the Sith Academy Entrance.
(Just as a note, you may eventually be able to buy items from Mika Dorin, the bartender in the cantina in Dreshdae. He sells some fantastically expensive items, such as +5 Strength gloves for more than 20,000 credits. He won't initially offer to sell you items; after you complete the Leviathan portion of the game, a mysterious man will approach you at one of the starports who will tell you to visit Mika; after that, you'll be able to purchase anything you like at this secret shop.)
Sith Academy Entrance
Objectives: Find a way into the Sith Academy
Speak to the guard at the main entrance. He'll inform you of the Sith medallion that you'll need to gain access to the Academy. If you haven't obtained one of these yet, return to the cantina and speak to Yuthura Ban, a Twi'lek Dark Jedi Master. Before you go back, though, you might want to talk to the starving Sith apprentices near the door; they're being slowly tormented by Mekel, who is refusing them food or water. If you speak to them, then speak to Mekel, you will have a chance of getting him to leave for dinner, if you choose the correct responses ("I won't let you do this!", etc.). Once he's gone, speak to each of the candidates again. You can attempt to get them to leave, or have them do something foolish, all of which requires successful persuasion, resulting in the corresponding shifts in Force alignment.
If you're playing Dark side, then you can simply accept Yuthura's offer of sponsorship into the Sith academy, and you'll be teleported there immediately. If not, however, you should attempt to lie to her to gain entrance. She won't fall for it, and will reject your application.
Head back up the ramp towards the Ebon Hawk. You'll be accosted by Sith hooligans, eager to kill seemingly just for the fun of it. You can accept their offer of a fight, or attempt to avoid a fight in order to gain Light side points. If you do kill them, one of their bodies will have the Sith medallion that you need.
If you don't kill them, or even if you do, you can proceed further back towards the Hawk to encounter yet another crazy Sith who wants to kill people. If you attempt to protect the Twi'lek he's threatening, he'll go hostile with you; kill him for a Sith medallion and a bondar crystal.
Once you have your Medallion, you can talk to some of the other prospective Sith to tease them about it, or just talk to Yuthura to gain access to the Academy.