Before you even get to take a breath of fresh air, Bastila will want you to come with her and visit the Jedi Council. Hope you don't have a lightsaber phobia, because it looks like you'll be the proud owner of one pretty soon.
Before you follow Bastila to the Jedi Council, speak with the Twi'lek in the courtyard to learn of a missing girl, apparently captured during a Mandalorian raid some time ago. This isn't a quest, yet; just some useful information to have. You can also visit the merchants off to the east of this landing pad and sell your excess equipment.
Once you reach the Jedi Council, you'll learn that they're not entirely sure of your true capabilities. They'll discuss the situation, and send you away while they deliberate, only to call you back the next morning. Make a party, but don't bother outfitting them yet. Once you arrive at the Jedi Council for the second time, you'll be invited to join the Jedi Order, after the Council judges your mental state. You can lie every time you are given that option to take a Dark side hit; fortunately, the Council is fairly gullible, and understandably so: it'd be a pretty short game for Dark side players if you got washed out before you even became a Jedi. A quick montage sequence will zoom you through the next few weeks of your PC's life. When you finish the montage, you'll be facing Zhar Lestin, the Jedi Master who has been instructed to test you to ensure your fidelity to the Force.
First off, return to the council hall and speak to Masters Vandar and Vrook to learn of the Jedi Code, which constitutes the first of the three tests you need to pass. (Master Dorak also has some fairly interesting backstory about Malak that you will probably want to hear.) Once you have learned the Code, return to Zhar to attempt the first test. (The answers, in order, are: peace, knowledge, serenity, harmony, and the Force.)
After you pass the first test, return to Dorak and learn a bit about lightsabers. He'll instruct you as to the meaning of the different colors of the blade, and eventually ask you what Jedi class you'd like to proceed as. Read over their definitions in the class section near the beginning of this guide to make sure you're clear on the differences between the classes; this is a binding choice! (In short, soldier = guardian, scout = sentinel, and scoundrel = consular. You can do a complete 180 from your initial class, if you wish, or continue along the path that you've been working on since the game began.)
Once you've chosen your class and gone through the level-up procedure, Dorak will give you a crystal and send you back to Zhar, who will help you construct your saber. Finally! Now you can proceed to the third and final quest; Zhar wants you to find the source of a "taint" that has been infecting a meditation grove near the Jedi Academy. If you need to remove items from some members of your party, go back to the Ebon Hawk and do so, then reform your party and outfit them appropriately. Head through the exit to the outer courtyard when you're ready to proceed with the quest.