A Doubting Sith: To the south of your starting point, you'll find Kel Algwinn, a Sith candidate who appears to be somewhat reticent about his career choice. If you can manage to persuade or Force persuade him to tell you what's wrong, he'll open up about his insecurities. Telling him to join the Jedi is a Light side choice; everything else is either neutral or Dark side, and you can earn a prestige point for eating cheese and ratting him out to Uthar.
Aiding Lashowe: Lashowe, the female candidate for the Sith position, has a bead on an artifact, the retrieval of which would probably impress Uthar to no end. If you can manage to persuade her into telling you what it is, and then letting you help her out, she'll relocate to the far northern end of the Valley of the Dark Lords.
The Hermit in the Hills: Speak to the Sith teacher watching over the lightsaber contestants near the door to the valley to pick up this quest. It seems that Uthar's former master has gone insane, and now inhabits the nearby hills, kidnapping Sith students. Killing him might gain you prestige.
The Mandalorian Weapons Cache: There's a Mandalorian being interrogated in the eastern wing of the Academy; if you ask the interrogator there for a go, he'll let you attempt to get the information that the Sith value, regarding the location of the Mandalorian's weapon stash. If you have high computer use skills, you can attempt to convince the Mandalorian that you intend to help him, then use the console to send him into a coma, from which he will recover, but be useless to the Sith while in.
If you want prestige, and a Dark side penalty, inject the Mandalorian with two small doses of truth serum, then one of the biggest doses, and only then ask him about the weapons. He'll dish with the info, which the interrogator will attempt to claim as the result of his work. Kill the interrogator, then return to Uthar to gain that prestige.
Rogue Droid: This quest isn't available until you reach the Valley of the Dark Lords. Speak to the Sith student outside the Tomb of Marko Ragnus to pick it up, then head inside the same tomb to complete the quest.
Say hello to one of the game's abusable loopholes.
One last area of interest here, while not quest-related, is the Dueling Room in the southeastern corner of the academy. While there's not much to do here, there is a computer terminal that you can slice into; one of its options allows you access to the academy's datafiles. Good hackers will able to repeatedly perform this action for very few spikes; T3-M4 is a good choice for this task, obviously, and if you can get his computer use skill up above 36 or so, you can repeatedly perform the action for free. The reason you'd want to do it over and over again is that the game erroneously awards you experience each time you do it, thus creating an experience well that's limited only by your available computer spikes, or by the game's experience cap, if you do manage to hack for free. Cheesy, yes, but very handy for power-leveling.
After you have completed enough of these quests to sufficiently impress Uthar, he'll congratulate you and let you know that you've won. Your final test occurs next, and you'll be going alone, so be sure to loot your teammates of the best lightsabers and other equipment. Once you talk to Uthar again, you'll be whisked off to the Tomb of Naga Sadow.