Joining the Circle of Water means proving yourself to 
Saturas and the other water mages in the New Camp, then passing a test in a flooded ruin off the coast. You talk to the mages, swim out to an underwater temple, and solve a water-balancing puzzle before coming back for the initiation itself. The mages keep to a routine, so timing your visit makes the first part much easier.
You need to speak with several mages in the New Camp before anything moves forward, and the easiest time to catch them together is in the morning, when they gather in the dining area for breakfast. The three who matter are 
Riordian, 
Merdarion, and Saturas.
Start by talking to the water mages in the New Camp. Saturas is the one you ultimately need, but speak to the others first. Head to the breakfast area in the morning to find them gathered in one place.
Find Riordian first and ask him to help you become an alchemist. Run through all of his dialogue. That covers one part of your preparation.
Next, find Merdarion. He is a bit prickly at first, but keep working through his dialogue. Ask him about scrolls, then ask him to teach you how to craft them, and exhaust everything he has to say.
With the other mages handled and Saturas' advice followed, go back to Saturas and tell him you are ready for the next stage. He will not spell out exactly where to go, so here it is: your destination is the shipwreck on the beach.
Travel to the beach with the shipwreck and head down onto the sand. There may be enemies around, and you can either fight them or just run past. Go left through the broken ship into the next stretch of beach, then follow it all the way to the edge.
At the water's edge, stand between the two rocks and swim straight out toward the rocks in the distance. The underwater ruins come into view ahead. Swim for the entrance; it does not matter whether you stay on the surface or go under until you reach the doorway.
Go through the doorway. Inside, you have to dive, so make sure your breath is topped up, then swim down through the door. You will feel a current, so follow it upward until you reach the stairs. Once you are in the shallows you can walk again, so carry on forward into the puzzle room.
At the back of the puzzle room is a body. Loot it if you want, since it carries a few useful items. Directly behind it is the lever that controls the water puzzle.
The puzzle is about water filling tubs around the room. As each tub fills it spins, and your goal is to end with the water balanced evenly, two full tubs on each side.
- Pull the lever behind the body. The floor shifts and the tubs around the edges start filling.
- Watch them fill one at a time. The moment the fourth tub begins to spin, pull the lever again to send the water to the other side.
- Let it settle. You want two full tubs on each side, and when the second tub on the far side fills and spins, the puzzle is done.
You know it worked when the room starts glowing blue.
Timing is the whole trick: pull the lever the second time exactly as the fourth tub starts to spin.
With the puzzle solved, fast travel back to the New Camp and find Saturas again. Tell him you finished the task.
After you speak with Saturas, a scripted initiation scene plays. Once it and the dialogue wrap up, Initiation into the Circle of Water is complete.
Run through the checklist before you call it done.
- Asked Riordian about becoming an alchemist.
- Asked Merdarion to teach you scroll crafting.
- Returned to Saturas and set out for the shipwreck beach.
- Swam to the underwater ruins and into the temple.
- Pulled the lever, then pulled it again as the fourth tub spun.
- Waited for the room to glow blue.
- Returned to Saturas for the initiation.