Objectives: Obtain Sith launch codes
Before you open the door here, make sure your entire party is healed. Trust us on this; this will be absolutely critical in the upcoming fight.
Through the door to the north is the Sith Governor; he's the first truly difficult boss fight you'll encounter. He's an absolute badass that can rip any member of your party apart, even Bastila; you'll need to coordinate your efforts in order to defeat him. Even without a lightsaber, he's tough, and quite capable of dealing 40+ damage on a critical hit.
Before you fight him, though, you might want to try laying down a bunch of mines in front of the door leading south. You may have a few in your inventory, if you had Mission recover the ones you encountered in the sewers, and minor mines only require a demolitions rank of one to set properly. If you open the door, then lure him back into the antechamber, you may wind up knocking half of his life off with just a few frag mines. It might be worth a try if you're having an overly difficult time with this boss.
If that fails, then believe it or not, T3-M4 may just wind up winning the battle for you (if he's in your party). Since the droid is immune to the Sith's stunning Force attack, the Sith won't be able to stop his attacks without killing him. That will occur soon enough, but not before you use your shiny new flamethrower to put a serious dent in the Sith's health. Even if the Sith makes his saving throw against the fire, he'll still take at least 15 damage each time the flamethrower is used, so queue it up three times and hope T3 lasts long enough get them all off. If he does, then you should be able to deal the rest of the damage with Bastila or another character fairly easily; the Sith's defense ranking is only 19, which is definitely high, but not impossibly so. If T3 isn't in your party, then you'll have to keep quite a careful eye on your vitality levels and simply wait him out.
Where does he get those wonderful toys?.
After that thriller, search the Sith's body for the launch codes, and don't miss the strong boxes that contain some more goodies. Return to the Upper City, then take the elevator downstairs to meet Canderous in the Lower City Cantina. After hooking up with him, you'll be whisked off to Davik's compound. Canderous will have to be one of your party members; choose the other wisely. If you're a scoundrel with high ranks in security and perhaps computer use, then Bastila is a good choice; there's going to be a lot of fighting in the next area. Otherwise, you may want to bring along Mission or T3-M4, if only to get around some locked doors.