Objectives: Escape the Endar Spire
The Endar Spire is intended to be Knights of the Old Republic's in-game tutorial; you'll learn all about the controls and options here, if you didn't bother to read the manual.
After your character awakens for the first time, you'll find yourself accosted by someone who barges into your room without even checking to see if you're dressed. That's pretty rude, even if the ship is under attack, but you'll have to forgive Trask for being an insensitive brute and take him into your party. Grab your equipment from the footlocker nearby and head out into the hallway.
Eventually, you'll run across a Republic soldier getting taken down by a couple of Sith. Trask will display his courage by pushing you out into the hall to take the Sith on by yourself, giving you merely a lame battle cry as support. Once the fight begins, you should cycle over to him and walk him out into the hallway; be sure both he and your PC target the same character. This is the most basic, but useful strategy in the game's many fights: pick a target and take him down before moving to the next one. Your characters are much better off when coordinating their attacks on a single powerful foe than by splitting up and attacking whomever the computer chooses for them to fight.
Luckily for you, the Sith troopers on the Endar Spire are completely inept; they have a huge penalty to both their attack and defense, so you shouldn't have too much trouble with them. If your health starts running low, however, make sure you use a medpac immediately; you don't have a large margin of error between being wounded and being dead.
Once the first two opponents are dead, head through the door nearby to meet up with two more foes; kill them and loot the room. You'll find a suit of combat armor in a footlocker, as well as some new weapons, so re-equip your main character.
Proceeding on, you'll run across a squad of five Sith troopers who have just wiped out a bunch of Republic soldiers. Run towards them a bit, to ensure that the reinforcements arrive, then throw a frag grenade at one of the soldiers in the middle of the group; this should take out most of them and severely weaken the survivors.
Continue through the ship, checking bodies for equipment and items. You'll eventually reach the bridge, where Trask will prevent you from opening the door unless you equip a melee weapon. (This is due to the fact that enemies with melee weapons will get a +10 bonus to their attack rolls if you are equipped with a blaster.) Take out everyone in the bridge, then head out to the starboard side of the Endar Spire.
Objectives: Meet with Carth Onasi at the escape pods
This brief section starts with a duel with a single Sith trooper. Even without Trask to help you out, you should be able to take him down easily. If you're a scoundrel, however, you'll be forced to activate stealth mode to bypass this guy; he seems to be drastically more powerful when you're a scoundrel than when you're any other class, so use your stealth field generator and leave him behind.
You have a couple of options to get through the room separating you and Carth: you can either repair the droid and send him in to attack the troopers, or use the computer terminal to overload a power conduit near the troopers, killing all of them instantly. The correct answer is: do both. You get experience for either feat, but you can get twice as much as you should if you first repair the droid, let it kill the troopers, then overload the conduit to explode into the now-empty room.
After all of the troopers go down, search the body of the red trooper for a prototype vibroblade, then head through the door to finally meet up with Carth. Take the escape pod to get away from the Endar Spire before it's destroyed.