Objectives: Win the Swoop Race
The Ithorian mechanic will explain the basics of swoop racing to you. Luckily, it isn't a very complicated mini-game; you use the analog stick to maneuver your bike to the left and right around obstacles, or onto accelerator pads, and you hit the A button once your bike is ready to upshift. The path that the bike travels is straight, so you won't have to worry too much about maneuvering, as long as you can avoid the small obstacles. Hitting the accelerator pads will be necessary to achieve an unbeatable time, and don't forget to upshift when you require extra speed.
The main trick here is to just suck on your first real run (the very first time you get on the track is a trial run); hit obstacles, don't accelerate, and so on. You don't have be outright terrible, but you'll want to get a time above 40 seconds for the first heat. The reason for this is that no matter what your time is, another racer will usually beat it after the first run, forcing you to race again; if you have a great first run, you might not be able to beat your own score, let alone the one set by your competitor. So take a dive on the first run, and save your best effort for the second heat.
After you achieve victory, a short cutscene plays out with Brejik and Bastila, culminating in a fight between yourself and a few of the Black Vulkars. Eliminate Brejik first, then his cronies. (If you have a hard time here, just run away from Brejik and let Bastila take care of the enemies.) After the fight, be sure to search Brejik's body before talking to Bastila; he's got a belt and an armband which are perfect for melee-weapon characters, as well as a set of gloves that give +1 to Dexterity. Oh, and did I mention the lightsaber? He's got one of those, too. Once you've rifled the bodies, talk to Bastila, which will automatically warp you back to your apartment.
Objectives: Find a way off this rock
Once you're done chatting with Bastila and Carth, leave the apartment with the party that you had before you finished the swoop race. You'll receive a tip that Canderous Ordo, a Mandalorian that works for Davik, is waiting for you in the Cantina. Unequip your other party members and head back inside, then leave again and choose the party that you want to use to finish off the rest of this planet's quests. Bastila is, quite obviously, an excellent choice; the other member should be a counterweight that possesses skills you're deficient in. If you do choose Bastila, she'll instantly gain three levels once you leave the apartment; load her up on skills that you think will come in handy, and take the chance to examine the new Jedi Feats and Force Powers. (Force Cure should be a mandatory choice for her level 6 Feat.) Head to the Cantina to meet up with Canderous.
Canderous has a grand scheme: he wants to send you into the military base that the Sith control in order to obtain the secret launch codes, without which any ship trying to take off will be destroyed by the Sith's orbital defense array. You'll first need to obtain the T3-M4 droid from Janice Nall; after your conversation with Canderous, Janice will finally offer to sell it to you. If you haven't yet, now might be a good time to earn some experience and credits in the Duel Ring. Even a scoundrel should be able to proceed far enough along to beat at least Ice, and now that you have Bastila to cure you, you won't have to waste medpacs on minor injuries between fights.
If you haven't come back to the Upper City in a while, you might also want to get the permacrete detonator for Matrik, inform Dia about the progress of her bounty quest, and give the Rakghoul serum to Zelka. Proceed to Janice in the North Upper City to get the droid; she'll want 2,000 credits for it, but you can persuade her down to 1,500 if you have enough points in that skill. If you're willing to take a Dark side hit, and have been seriously pumping persuasion, you can knock down the price to 1,000 credits, or even force her to hand over the droid for free.
T3-M4 will be a required member of your party for the next section of the quest. While you're leveling him up, you will probably want to pump his repair and computer use skills, as well as give him another rank in gear head, since all of this put together should get you up to at least the 16th rank in each skill, which will save you spikes and parts over the long term. You probably haven't seen Bastila in action yet, so you might want to take her along, as well.
Once you have your party set up, head over to the military base, near the elevator leading to the Lower City.