Divine Beast Vah Ruta was piloted by Mipha of the Zora, so you should seek out the Zora people.
Go to Lanayru Tower
After completing the Seek Out Impa quest, if you go into your Adventure Log and select the Free the Divine Beasts, you will see that one of the destinations is to the east. If you want to get the map of the region where that destination is located, you should start by warping to Kakariko Village. Then go northeast, and you will reach a fork in the path where the right side slopes upward, and the left side goes straight ahead. Go left at that fork and continue going northeast.
You will reach another fork in the road, but don't go near the traveler. Secretly he is a member of the Yiga Clan and will try to kill you. In general, you might want to avoid travelers who seem to be trying to get your attention, because they might be from the Yiga Clan. Go north at this fork, and soon you will see some smoke in the distance. Go toward it and you will find Eryck sitting by the camp fire. You can talk to him to learn more about Zora's Domain and the Lanayru Tower.
Go north from Eryck and you will soon see the Lanayru Tower. If you go straight toward the tower, you will soon find the Rutala River blocking your way. However, you can float down to an island that sticks out from the opposite side of the river, so do that. Go northwest along this island and you will pass by a sleeping Hinox. If you wake it up, you will have to fight it (or run away). It can pull up trees and use them as weapons, so be careful.
Northeast of the Hinox, there is a wooden bridge. Go across it. As you go across, a Zora named Tona will ask you to go to the nearby Inogo Bridge, where Prince Sidon is waiting in hopes of finding a Hylian to help him.
After you are done talking to Tona, go east on the bridge. From here, you can make your way up the slope to the Lanayru Tower. You can try to climb up the north side of this hill, but it will be constantly rainy on that side. If you make your way up the grassy slopes on the south side instead of trying to climb the rocks, you will encounter Bokoblins, Lizalfos, and Moblins on your way up. If you have arrows, you can shoot at the red Xs on the wooden posts that hold up the wooden platforms that some of the monsters are standing on. They can't swim, so they will drown if you drop them in water.
At the top of the tower, you will find a Zora named Gruve, who also asks you to meet Prince Sidon on the Inogo Bridge. Gruve will show you the bridge, and you can also see a shrine near that bridge. Be sure to activate this tower before you leave.
Soh Kofi Shrine: A Minor Test of Strength
After you activate the tower and get the map of the Lanayru Region, float down to the shrine north of the tower. Open it to add it as a travel gate. If you wish, you can go into the shrine.
This shrine is one of many "minor test of strength" shrines that you can find across the land. These contain a Guardian Scout I or Guardian Scout II. It might shoot beams at you as you approach, so raise your shield. If you have good timing, you can press A to use your shield to bounce the beam back at it, damaging it. Then go up to it and try to hit it. If its shield is raised, you can knock it back with a club or other heavy weapon. Otherwise, try to jump out of the way of its attacks at just the right moment so you can do a Flurry Rush.
If it jumps back and puts its weapons out at its sides, run behind a column (but don't stand too close to the column) because the Guardian will spin until it hits either you or some other obstacle. If it hits the column, it will be damaged and stunned for a while, so you can run up and attack many times.
If it jumps back and lowers its head, its about to spin around shooting a weak laser, so you can either try to rush up to it and hit it repeately to try to stop this attack, or you can run to a safe distance while it spins the laser. The laser creates an updraft, so you can jump and use the paraglider to float up, allowing you to shoot an arrow in slow motion at the Guardian.
Another attack it can do is charge up a powerful blast. It will jump back and raise its head before it does this. It is best if you can quickly rush up and attack it repeatedly in the hope of stopping it before it can charge up the attack. Otherwise, you will have to run out of the way of these blasts, because they will hurt you badly if they hit you.
When the Guardian is defeated, take its weapons/shields (which are useful against other Guardians) and get any Ancient Loot it drops, then go open the treasure chest and talk to Soh Kofi to get a Shrine Orb.
Meet Prince Sidon
From Soh Kofi Shrine, go east along the road to reach Inogo Bridge. When you get there, there will be a cut scene where you meet Prince Sidon. He says that the Divine Beast Vah Ruta has been causing heavy rainfall and asks you to help He gives you an Electro Elixir to help against the electrified enemies along the road to Zora's Domain. This starts the Reach Zora's Domain quest. After that, just follow the road as marked on the map. Sidon will talk to you at various points along the way. He will warn you of an approaching monster when you cross Luto's Crossing. Defeat the Moblin and continue along the path, and you will soon reach Zora's Domain. Sidon will ask you to go meet the king with him. Before you do, you might want to go to the lower level of Zora's Domain and open the shrine that you find there. You can also complete the shrine if you wish.
Ne'ez Yohma Shrine: Pushing Power
In this shrine, there is a slope with water running down it, and a bunch of stones rolling down the slope. First, go up the slope, being careful to avoid getting hit by a stone. At the top, you will find a glowing ball, as well as some raised walkways that you can use to walk above the watery slope. Walk onto the walkway to reach the treasure chest farther down the slope that is on top of a triangular platform. It contains a Zora Spear.
Then go back to the top of the watery slope. Use Cryonis to make a sideways ice pillar behind the orange ball to push it forward onto the watery slope. It will stop in front of a black stone.
Now use Cryonis to push away any stone boulders that are still on the watery slope so they won't get in the way of the orange ball. After that, shatter all the ice pillars that you have made.
Then go to where a laser is hitting a platform farther down the slope. You need to put an ice pillar in the gap near where the laser is hitting. That way, when the orange ball rolls down the triangular platform, it won't fall through the gap and roll off at the bottom; instead, it will roll to the second triangular platform that the laser is hitting.
Next, make a pillar after the second triangular platform to prevent the orange ball from rolling off of the side of the watery slope after it rolls down the second triangular platform.
When those two pillars are in place, go back to the orange ball and create an ice pillar behind it that pushes it up onto the stone platform. If you put the other pillars in the correct places, the orange ball will roll to the first triangular platform, then will hit the ice pillar and go to the second triangular platform, and from there it will hit the second ice pillar that you made, stopping at a platform near the target hole at the bottom of the ramp. Then you can just create an ice pillar under the orange ball to push it up onto the platform, where it can roll into the target hole, opening the gate. Go talk to Ne'ez Yohma to get the Shrine Orb.
Meet the King
Back in Zora's Domain, go up the large staircase to reach the throne room. This completes the Reach Zora's Domain quest. There will be a cut scene where you learn that Vah Ruta's endless rain will make the east reservoir overflow, destroying Zora's Domain and any Hylians living nearby. The Zoras have learned that Shock Arrows are effective against Vah Ruta, but because of the Zoras' aquatic nature, they can't safely touch shock arrows, so they need a Hylian's help. The king gives you Zora Armor that Mipha herself made. However, Muzu blames the Hylians for the Calamity and for Mipha's death, so he refuses to tell you how to get shock arrows.
Go back down to the Mipha statue where you found Sidon earlier. Talk to Muzu or Sidon. After a cut scene, Link recalls a memory of Mipha. Muzu doesn't believe Link, so equip the Zora Armor that Mipha made for Link. When Muzu sees that it fits Link perfectly (which it wouldn't if it had been made to fit a Zora), Muzu finally believes you.
Get Shock Arrows
He says to go up to Ploymus Mountain, where a Lynel shoots shock arrows. You will need 20 of them. Sidon says to go up the waterfall east of Zora's Domain to get to the mountain. There is a bridge to this waterfall near the throne room. The waterfall is in Mikau Lake. Be sure that your Zora Armor is equipped, then go into that waterfall and press A to swim up it. You automatically use the paraglider after you jump at the top of the waterfall. Glide to the next section of waterfall and swim up that one as well.
You will be near Lulu Lake. Look for shock arrows in this area. Take a picture of one for your Hyrule Compendium. You might want to target shock arrows so you can find them more easily in this area, because they are scattered somewhat randomly.
If you don't find shock arrows here, go up the waterfall in Lulu Lake and then look for shock arrows in the area at the top. Then carefully go up the mountain. There will be a cut scene when you approach the clearing, and Link will hide from the Lynel.
The Lynel has very good hearing and a good sense of smell, so when you hide from the Lynel, make sure to stay far away from it at all times. It also has good eyesight, so do your best to hide behind things if the Lynel is looking toward your location. The Lynel has a very strong bow, so if he hits you with even one arrow, you might die instantly. If the Lynel sees you and starts shooting, you might want to just warp back to Zora's Domain and come back to this area again.
Carefully collect 20 shock arrows from this area. They will be stuck in trees and rocks. You will need to go up to Shatterback Point afterward. There are four shock arrows at Shatterback Point, so you only really need to get 16 shock arrows in the area with the Lynel.
Open the Divine Beast
After you have your 20 shock arrows, go to Shatterback Point and glide off of it toward the place marked on your map. When you land on that pier in the reservoir, switch to the Cryonis rune, make sure your shock arrows are equipped, and equip the Zora Armor. You might also want to change your Camera Sensitivity setting (in Options) to Very Fast.
When you are ready, talk to Sidon and he will tell you the plan.
When Sidon swims to a waterfall, press A to swim up it. As you go up, turn the camera so it is facing the divine beast. When you get in the air above the divine beast, turn the camera to look at the pink orb at the top of the waterfall. When you take out your paraglider, shoot a shock arrow at the pink orb. You will aim in slow motion, but this takes stamina, so don't take too long.
After shooting the orb, you don't need to use your paraglider to float down. You can fall into the water and you won't get hurt. Then press A to hold onto Sidon, and immediately press L to start the Cryonis rune. Aim the camera at the divine beast so you can see the blocks of ice. Use the right control stick to aim at whichever ice block is coming toward you (using motion controls to fine-tune your aim if you have motion controls enabled) and press A to shatter the ice block before it hits you.
Once all of the ice blocks are gone, Sidon will swim to the waterfalls again. If possible, try to look at the divine best before Sidon gets too close so you can see which waterfalls still have a pink orb over them. This way, you won't go up a waterfall that you have already gone up.
Repeat this process until all pink orbs have been shot. Vah Ruta will open its door and Sidon will take you there. You will automatically register Vah Ruta as a travel gate, so feel free to warp away if you need to heal up. You can just warp back here when you are ready.