While in most Zelda titles, Zora Domain is one of the later regions that Link will explore in Hyrule, in Breath of the Wild, it is generally the first region that players will explore instead. As the only region that isn't burning hot or freezing cold, alongside the relatively weak enemies in the immediate area and the prize for clearing Vah Ruta, Mipha's Grace, being so useful, there's no reason to not go to Zora Domain before anywhere else.
Easily one of the largest cities in Breath of the Wild, Zora Domain has a ton of things to see and do within its walls without taking into account the Divine Beast and its associated quests, mostly due to how sprawling the area is to explore.
Merchants
Unlike in other cities, the merchants in Zora Domain are all out in the open and are fairly easy to spot. On the ground floor, just up the stairs from the Shrine in the center of town, players can find Marot's Mart, a combination store that holds the Coral Reef, a general store, as well as a workshop called the Hammerhead. The Coral Reef has several useful food items, such as Tabantha Wheat and Rock Salt, as well as a large amount of Ice Arrows, while the Hammerhead is the location where players can recreate Mipha's signature weapon, the Lightscale Trident, by handing over a Zora Spear, 5 Flint, and 1 Diamond. Players can also reforge a weakened variant on the Lightscale Trident, the Ceremonial Trident, which looks identical though does slightly less damage and requires a Zora Spear and 5 Flint to be given to the same NPC.
On the opposite side of the Shrine, players can also find a cooking pot in the first room while the Seabed Inn is found in the rear. For a mere 20 Rupees, Link can rest and regain all of his hearts, while for 80 Rupees, he will be able to rest on a water bed that will not only replenish all of his health but grant him three temporary hearts and one wheel of temporary stamina as well. Unfortunately, there is no armor shop in Zora Domain so that players can buy the Zora Armor. Instead, they will have to obtain the pieces, either by finding them or by completing certain side quests in order to obtain them.
Loot and Korok Seeds
For an otherwise peaceful city, there seems to be quite a few weapons lying around Zora Domain. In the room with the Ne'ez Yohma Shrine, players can find a Silver Longsword that respawns after every Blood Moon, a great weapon that players can consistently go back to if they need to stock up. On the upper branching levels, players can also find a Zora Sword on one side and a Zora Shield on the other, though they aren't particularly powerful without the assistance of attack boosting armor or food items.
While there are many Korok Seeds in the Lanayru region, there are far fewer inside Zora Domain itself, with only two being available inside the city walls. The first can be found by climbing up to the top of the throne room and onto the fish's head, being a simple interactable Korok, though reaching this can be incredibly difficult if the player hasn't defeated Vah Ruta and stopped the torrential rain yet, as Link won't simply be able to climb up the side of the building. The second Korok is in the same general location, though it found much higher up on the fish's tail instead. Unlike the first Korok, which can theoretically be reached by boosting up a waterfall and gliding towards it, the second Seed will more than likely require Vah Ruta to be completed first.
Quests
Aside from the Reach Zora Domain and Divine Beast Vah Ruta main quests, there are also eight side quests that can be completed in Zora Domain, one of which being quite useful for obtaining Diamonds so that players can reforge the Champion's signature weapons. This particular quest is given by Ledo, who can be found at the entrance to Zora Domain right after crossing the bridge. They will request that Link bring them back 10 Luminous Stones in exchange for Diamonds and is one of the easier ways to obtain Diamonds in Breath of the Wild. After giving Ledo the first 10 stones, they will reward Link with 2 Diamonds, then give him 1 Diamond for every 10 Luminous Stones they present to Ledo.
Another quest that is somewhat longwinded, but great for getting Rupees early in the game, is Special Delivery, which can be started by talking to Finley. The young Zora wants Link to help them deliver a love letter downstream and tasks Link with ensuring the letter reaches its intended recipient, which involves the player fighting off monsters to ensure the parcel arrives in one piece. If the letter is damaged and breaks, Link can return to Finley to get another copy of the letter and try again. After the letter is delivered, fast travel back to Zora Domain and meet with Finley to be given a Gold Rupee.
One side quest that should be grabbed as soon as the player reaches Zora Domain is Lynel Safari, given by Laflat found near the general store. They will request that Link bring them back a photo of the Lynel that lurks around the Lanayru region, the same Lynel that Link needs to sneak around to gather Shock Arrows for the main quest to be completed. Returning with a photo of the Lynel will reward Link with the Zora Greaves and is the only way to obtain this armor piece in the game.
Another monster related side quest can be obtained in Zora Domain, this time involving a Hinox that roams Ralis Pond. After speaking with Torfeau, they will ask Link to hunt down and kill the monster, which they will reward him with a Silver Rupee should he be successful.
One of the more annoying side quests can be completed on the road up to Zora Domain and requires the player to read all of the stone tablets that dictate the history of the Zora. There are ten of these tablets in total, the majority of which being fairly easy to spot if players stick to the main path up to the city. After all 10 have been read, talk to Jihato, who is standing in front of a miniature tablet, to be rewarded with a Diamond.
Among the easier side quests, Link can speak to a Zora child, Tumbo, who is running around the area between the inn and the general store, who is looking to collect and sell Hot Footed Frogs to make money while his mother is missing. These frogs are very easy to spot and catch on the road up to Zora Domain, so this is one that can be completed as soon as the player reaches the city. After presenting the child with 5 frogs, they will give Link an Armoranth as thanks.
On the upper level, in the pools area, there is a Zora named Fronk who is worried for his wife's safety, as she hasn't returned home yet and he has heard no word about her since she left. Travel to Lake Hylia to find her swimming around without a care in the world among several smaller islands to the west of Hyrule Bridge. After realizing that she is going to be in trouble with her husband, she gifts Link with 5 Staminoka Bass and races back to Zora Domain.
The final side quest in Zora Domain is a fairly simple one, but one that requires the player to have obtained the Zora Armor from King Dorephan first. After talking to Gruve, who is standing next to a small waterfall, he will ask Link to dive off of the waterfall, then swim his way back up using the power of the Zora Armor. Once this has been done, Link will be rewarded with 5 Fleet-Lotus Seeds.
Shrines
While the Lanayru area is extensive, there are only two Shrines within the Zora Domain region. The first is the Ne'ez Yohma Shrine, which can be entered as soon as players reach the city, is a simple Cryonis puzzle that requires Link to maneuver a Shrine Orb into position around a maze-like structure. The second, Dagah Keek Shrine, is part of a Shrine quest called The Ceremonial Song, which can be started by talking to a young Zora named Laruta, who is usually standing in front of Mipha's statue in the center of Zora Domain. In this quest, you will need to track down the Ceremonial Trident, which can be obtained by talking to Trello to find out where he left it. Once taken, go to the marked location near the waterfall and swim up to get some height, then do a diving attack with the trident equipped onto the Shrine pedestal to reveal the Dagah Keek Shrine. Once inside, Link simply needs to approach the Monk to receive the Spirit Orb.