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If you want to join the Inquisition, you'll first have to speak with Inquisitor Raphael in the Inquisition Foyer. He'll want to test your mettle, so he'll give you an introductory quest to assist Inquisitor Dominguez in the Temple District. Once you do that and return, he'll ask you to tithe 1000 gold (or 1500 gold if you're tainted) to prove your dedication. Once you do that as well (250 experience) you'll become an Acolyte, which gives you a bonus of +5 HP, +5 unarmed melee, and +5 cold / fire / electrical resistance.

Then Raphael will give you the following quests:

1. Deal with the Afflicted - This means you have to kill all the wererats (beggars) in the sewers, including Enrique the beggar leader. If you entered the sewers from the Port District, then you also need to kill the beggar who asked you to choose sides. So if you're working your way through the Beggar faction quests, you should do them before completing this quest, or else you won't receive the final Beggar award. Your quest log will update when you've killed the last wererat, so that will tell you when you've found them all. Then report your success to Raphael (750 experience, +25 karma).

2. Create a Rod of the Inquisitor - First you need to pick up some darkwood, so head out to El Bosque and talk to the woodcutter there. He'll tell you he has darkwood in his basement, but that he can't get to it currently because undead have taken over the place. So kill the undead for him, and he'll give you the darkwood you need (750 experience). Then report back to Raphael (500 experience), and he'll tell you that you need the Hair of a Saint. So go to the cathedral. If the knight out front stops you, just say that Raphael sent you. Inside, go to the far side of the room and enter the crypt. You can get the hair you need just by clicking on the coffin. Then when you return to Raphael (500 experience), he'll tell you that the Grand Inquisitor must bless the rod before it is complete. So go downstairs and get the blessing, and you'll receive the Rod of the Inquisitor (+1 perception, +1 healing rate, +10 to your Divine Favor branch).

3. Rescue Inquisitor Darsh from the Goblin Camp - Just escort Darsh from the Goblin Camp to the Barcelona Gate District, where he'll give you the Amulet of the Grand Master (+10 speech, +2 AC, +1 healing rate) and you'll receive 750 experience. Then return to Raphael to complete the quest (1000 experience, 25 karma). You can still complete the quest even if Darsh dies.

4. Locate La Calle Perdida - If you haven't found the Wielder hideout yet, then go talk to Quinn in the Gate District, and follow events from there. Once you know the location, you'll have a choice: you can lie and say it doesn't exist, or you can tell the Inquisition where it is. If you lie, then you'll simply gain 2000 experience. If you tell the truth, then you'll gain 2000 experience plus 50 karma plus receive the Hero of the Inquisition title perk. But all the Wielders will disappear, and you won't be able to use La Calle Perdida's Way Station any more.

Completing the final quest will make you an Inquisitor, which gives you a (total) bonus of +5 HP, +1 perception, +10 unarmed, +1 skill point per level, and +10 fire / cold / electrical resistance.

Finally, you'll receive a quest to protect the Crown of Thorns at Montserrat, but that quest is the same for all factions.