Objectives: Find Bastila
Welcome to Taris, the first planet you encounter in Knights of the Old Republic. When you wake up from your dream, Carth is there to provide some handy exposition. After your escape pod crashed on Taris, he luckily managed to find a luxury apartment that just happened to be abandoned by its previous owners. (I suspect foul play; how much do you know about this guy, anyway?) This apartment will be your base of operations for the duration of your stay on the planet; you can warp back here almost any time you feel like it using the map screen in the options menu.
As soon as you exit your apartment, you witness an altercation between a Sith patrol and some aliens. The Sith, of course, spot you, and you find yourself in a fight before you know it. Luckily, you have a bit of an edge in the ion blaster you found on the red trooper's body back on the Endar Spire; it does an extra 1-10 damage when it hits a droid, and the Sith patrol just happens to consist of two battle droids. If you're not a blaster user, let Carth equip the ion blaster, and then go to town. The troops go down easily enough, and you should gain enough experience to hit another level-up.
Explore the Apartments if you wish; you can use Carth to unlock the doors if you don't have points in the security skill. You can steal anything you want from these people's lockers and storage containers; don't worry, petty theft apparently isn't much of an issue to the Force, so larceny won't start you on the path to the Dark side. There is a side-quest here involving a woman named Dia; see the Side Quests at the end of this planet's section to learn more. You'll also run into a merchant named Larrim here; you may want to take this opportunity to familiarize yourself with the merchant interface and sell off any unneeded equipment. Don't bother buying anything, though; he charges 15% more than another merchant you'll encounter in the next area. Once you're done running around, head through the doors to the south that lead to the Upper City.
Objectives: Find Bastila
Go ahead and look around for a bit; there's plenty to do and see on Taris. Most of this is optional, but you can earn credits, find out information, discover Side Quests, and perhaps take your first steps towards the Light or Dark sides of the Force by interacting with the residents of Taris. (In case you haven't figured it out yet, NPCs with proper names will generally have something of interest to say to you; generic characters like "Taris Citizen" will have a few canned responses to your attempts at conversation.) You can stop by Kebla Yurt's shop for reasonably-priced goods, or head over to Zelka Forn's medical facility for healing, if you require any. (Healing from NPCs is free.) One event of note is the bounty hunter incident near the entrance to the North Upper City. A merchant who owes protection fees is being hounded by bounty hunters looking to collect on the debt. They eventually decide to make an example of him. You can step in if you like; the choices that follow will determine whether or not you take a step towards the Light side or the Dark side.
There are some other areas of interest in the Upper City, so once you're done looking around... take a look around elsewhere. There are two different ways to prod on the main quest of finding Bastila. In short, you'll need to travel to the Under City, but in order to do so, you'll need to find a way past the Sith Trooper that guards the only elevator leading down. In a word: disguise.
Method #1: The first, and perhaps most obscure method of obtaining the required Sith uniform, is found in the Cantina, which is located directly north of the exit from the South Apartments. There are a couple of off-duty Sith there named Yun Genda (the man) and Sarna (the woman). If you talk to the Sith that's of the opposite sex of your character, you can get invited back to a party that they're throwing later that night. You don't need any points in persuasion; if you simply choose the right responses ("Oh, I'm not from around here." "Don't you need to blow off steam every once in a while?" "There's a bucket of fried chicken in my pants--wanna get greasy?" etc., etc.) you can finagle an invite without needing to persuade the character at all. Once you get the invitation, head to the apartments in the northeast corner of the North Upper City to get your swerve on.
Once you arrive, Yun and Sarna will chat for a bit, then everyone will pass out from too much ale. Apparently the Sith can't hold their ale--has our intrepid hero found the key to ending this horrible conflict once and for all?!? Well, no--but you do find some Sith armor in a Sith backpack on the floor near the unconscious partygoers.
Method #2: If you hit the Upper North City apartments before you talk to the Sith in the Cantina, or if you manage to anger your opposite-sex character, then when you arrive at the same location as the party, you'll instead find a few Sith in full regalia pestering an alien. Kill all of them, and you'll find some Sith armor on one of their bodies.
Once you find the suit, head to the Lower City elevator to the west of the apartments, equip the armor, and head down below. Be sure to talk to the guard before going through the elevator for a tidy experience bonus.