Joining an Order with in the city should not be done lightly. Take time to consider where you see your ambitions lived out--where your skills and interests pull you. When you have made your choice, write the next entry to your hero's journey (Fear and Refusal & Mentor) about joining an Order.
Cleric
Clerics are the few who would rise up in the name of Nira and cast the world's magicks in her name. They work specifically for the will of Nira, acting in her name to heal the sick and help the needy. In times of great need, they will rise up even in the face of great danger to protect what Nira would have them protect.

To be a cleric, simply declare your intentions to a clergy member with an oath to your duties in Nira's name.
"For it is us who shall rise in the face of turmoil and right it by the will of Her."
You will find the clerics in the great cathedral. Talk with a Priest or Priestess of Nira to give your oath.
Templar
Templars are the many who would stand and fight against the foes who would come to take Elantris for themselves. An army of the divine, motivated by their new home and its keeper. Swordsmen, mages, and any who would place themselves between enemies and the sanctity of the holy city and its residents.

No experience necessary, the only requirement for joining the templars is the will to fight with devout constitution in the name of a new homeland and for those who could not fight for it themselves.
"The enemy is found where benevolence, hope, and love are blotted out. Where the lives of my brothers are thwarted by darkness, that is where you will find me standing."
You can find the templars in the city's meeting hall. Speak with Commander Tremrith Slatvizla to commit yourself to the Order of the Templar.
Inquisitor
Inquisitors are specialists of sorts. There role, simply put, is to shine the light where darkness lurks. Masters of monsters and dark magics, they use any means to protect the greater city and its people from darkness itself. Many powerful and menacing creatures are attracted to the allure of Elantris, and many horrid things lurk in the Mistborn Forest just outside the city walls.

Witch hunters, monster slayers---they go by many names, but all are devoted to the extermination of dark magics and their beasts.
"Do you think the darkness scares me, beast? I am the darkness, and so, I am your master."
The inquisitors can be found stationed in the high tower libraries, but be sure to bring evidence of your desire to join them. Perhaps you should visit the Mistborn Forest first, and bring back a token of your potential within the Order. Speak with the Lord Inquisitor, Javilar Caarksa when you are ready.