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The Legend of Zelda: Tears of the Kingdom isn't known for being linear. Once you complete the main first quest, you are able to travel around Hyrule and do whatever you want to do. This can make figuring out the best course of action pretty difficult.
You will be tasked with visiting 4 different peoples of Hyrule: the Zora, the Rito, the Gerudo, and the Goron. However, the game doesn't outright tell you the best way to complete these 4. This guide will explain the easiest way to do things.
Updated by Erin Rice on May 29, 2023: After completing the game, it is obvious that there are more dungeons than originally thought when the game first came out. This guide includes the final dungeon, Hyrule Castle Chasm. It also added more information on how to get to the Spirit Temple within The Depths.
Wind Temple
When you are speaking to Purah in Lookout Landing toward the beginning of the game, she will allude to the fact that Rito Village is the best place to go first. She will recommend that you go speak to the Lucky Clover Gazette, which is located near Rito Village. The Wind Temple itself is the easiest of all the Temples available.
You don't need anything special and can head there as soon as you're given free rein to Hyrule. As long as you have a few food items, you should be able to complete this Temple with ease. If you get stuck, you can find a guide to the Wind Temple here.
Water Temple
The Water Temple can be accessed by completing the main quest for Zora's Domain. There isn't much of a difference between The Water Temple and The Wind Temple. The reason this Temple is recommended to be finished after The Wind Temple is due to the fact that the boss within the Temple is not nearly as easy as The Wind Temple's boss. In addition, Sidon's ability isn't nearly as impressive as Tulin's ability.
Make sure to stock up on plenty of Splash Fruit or Chuchu Jelly; that way, you can cleanse the sludge that is all over the Temple.
If you need help, you can find a guide to the Temple here.
Fire Temple
The Fire Temple can be accessed by completing the main quest of Goron City, and it's the Temple where things can get pretty challenging. Since you did The Water Temple first, Sidon's water power is invaluable in this Temple. Water can easily destroy any of the enemies in The Fire Temple. The water shield can also protect you from the many different attacks from the enemies within.
The ability you get from The Fire Temple allows you to break rocks, which can also be extremely useful. Depending on how much you value this skill, you could do this before The Water Temple. You can find a guide to the Fire Temple here.
Lightning Temple
No matter which path you follow, it might be best to save Gerudo Town and the Lightning Temple for last — mainly due to the fact this can be the most challenging Temple. The ability that you get from the Temple can be helpful but isn't nearly as good as the other abilities that you can get from the different Temples.
You want to make sure that you have plenty of lightning resistance before trying to take this Temple on. You can find a guide to the Lightning Temple here.
Hyrule Castle
This final dungeon, Hyrule Castle, is the hardest dungeon by far. You will experience the inside of the castle and have your final showdown with the Demon King, Ganondorf. You will fight Phantom Ganon here and need to stock up on a ton of Gloom healing items before coming. Ganon will be able to poison you with Gloom. If you want, you can pick up The Depths set, which will give you extra Hearts to lose to Gloom before your own. You can find a guide to defeating Phantom Ganon here.
Spirit Temple
This is a dungeon that you might not know about at all. It is a secret dungeon that is found within The Depths. However, you need to go to Dragonhead Island to get the quest to come to this dungeon. This dungeon is extremely difficult. You don't want to try to do this dungeon until you have a good amount of health. You also will want some good weapons, upgraded armor, and plenty of food. You will need to complete a pretty large quest first in order to start the Temple. This quest is called "Guidance of Ages Past" and can be obtained after exploring the Ring Ruins and completing Dragonhead Island. You will reach the island when completing the quest that gives you the Charged Armor set. You will go there during a pretty important quest for the main story.
Hyrule Castle Chasm
After you find the fifth Sage, she will be tasked with defeating Ganon, the Demon King, once and for all. You'll be told he has entered into The Depths, and after defeating the Yiga Clan there, you will learn that he is in the chasm underneath Hyrule Castle. This is the final dungeon of the game. You must defeat Ganon in an epic conclusion. You will need plenty of food, good weapons, good armor (you may want The Depths Armor set), and lots of resources. This is the final confrontation of the game.