Objectives: Complete the third Jedi Test
In the Courtyard, you'll run across a few named NPCs. One of them, named Jon, will interrupt your travel and request that you hunt down and eliminate the Mandalorians that killed his daughter; you'll be able to pursue this quest soon enough. Elise has another quest for you; her droid has gone missing, and she wants you to find it. All in good time.
Proceeding south from Elise, you'll run into a few of the previously mentioned Kath hounds. These guys are very weak, so take this opportunity to slice-n-dice with your newfound lightsaber. You can wander through this area, killing all of the Kath hounds that you find. When you're done, take the westernmost path to the Matale grounds.
Matale Grounds
Objectives: Find the source of the dark taint
Take a look around once you arrive in the Matale Grounds; you should see a group of people standing around some kind of transport on a hill near you. Walk towards them to witness a typical Mandalorian solution to a problem. After the Mandalorian eliminates the settler, he will turn on you and become hostile. If you're lucky, the Duros nearby won't notice the fighting and will leave you be for now, so gang up on the Mandalorian and take him down, then finish the other gangsters off. Don't worry about the rest of this area yet; continue south into the Grove.
Grove
Objectives: Find the source of the dark taint
The grove is a smallish area, but with another side quest implanted in it. See the Murdered Settler side quest entry for more info.
Besides that, there are more Kath hounds around, as well as a group of Mandalorians. In a clearing far on the east side of the area, you'll find the source of the darkness; apparently a Dark Jedi named Juhani has set up camp in a small graveyard-like area. When you approach, she'll declare herself to be your destroyer, incapacitate your party members, and engage you in a one-on-one duel. Whip out your lightsaber, and get it on.
If you have enough vitality, this fight shouldn't pose much of a threat to you, but if you are having trouble, you might want to activate one of those energy shields you should have lying around in your inventory--it'll help blunt the damage from her saber a bit. She deals a fair amount of damage, but doesn't do so very quickly, so you should be able to stay on top of things with cure or medpacs.
You'll want to keep Juhani alive; she'll be a valuable party member later on.
Once you knock her life down far enough, she'll surrender and offer you a host of dialog options; follow the quest through to its end according to your character's personality. Your best option is to attempt to persuade or talk her into returning to the Jedi Order, since she will be available as a party member eventually if you manage to convince her to repent. You can kill her if you wish, but even Dark Jedi should recognize the usefulness of having another Jedi in the party; what's more, if you do bring her into your party, you'll eventually be able to shift her back over to the Dark side. Once you've disposed of Juhani, however you choose to do so, return to the Jedi Council and speak to Zhar. You should also visit with Juhani if you did manage to bring her back to the Light side of the force, but you won't be able to add her to your party yet.
Once you return to Zhar, you will be officially inducted into the Jedi Order and given a set of Jedi robes. Speak to Master Vandar to learn about your next main quest, which is to investigate the mysterious ruins which lie to the east of the Courtyard. Afterwards, the paterfamilias of the Matale family will barge into the chambers, and accuse the Sandral family of kidnapping his son. This opens up the Sandral-Matale Feud side quest.