After an hour or two of tutorials and exploration, The Legend of Zelda: Breath of the Wild lays out the stakes in pretty clear terms: Calamity Ganon has corrupted four divine beasts, which are necessary tools to defeating him. It's up to Link to visit each divine beast and free them from Ganon's corruption. This, in essence, is the main story of Breath of the Wild.
Warning: It should be pretty obvious by now, but there are spoilers in the article. Turn back now if you want to avoid them.
Like anything else in Breath of the Wild, you can tackle the divine beasts at your leisure, visiting them in any order you want at any time you want. If you want to finish all 120 shrines and get the Master Sword before cleansing a single divine beast, nobody is stopping you. You can even hightail it to Hyrule Castle and defeat Ganon without doing any of that, though you'll be missing out on plenty of fun stuff and cool powers.
Zelda: Breath of the Wild Divine Beasts Order: What's the best way to do it?
The divine beasts are this game's equivalent of dungeons from past Zelda games. They're four humungous mechanical animals in which you'll run around inside to solve puzzles, defeating a boss at the end of each one. You'll also get a special, very valuable power for finishing each of them.
The giant elephant Vah Ruta is an easy first choice, as it's closest to the place where you initially get the quest to cleanse the beasts. Follow your waypoint north to Zora's Domain, which you can't miss because there are about a million Zoras on the way who will annoyingly beg for you to go there. The lead-up to the dungeon is fairly straightforward and the dungeon itself isn't especially difficult. Best of all, you get Mipha's Grace for finishing it, which will automatically resurrect you with full health (and extra hearts) when you die. It has a lengthy cooldown, but this will be invaluable for the rest of your journey.
From here, it's really up to you. Since the game is designed to give you this freedom, the dungeons don't necessarily scale upward in difficulty. I did Vah Medoh in the northwest side of the map second, followed by Vah Naboris in the southwestern desert and then Vah Rudania in the northeast. I will personally recommend doing Vah Rudania last since the ability you get is probably the least useful of the bunch. Daruk's Protection acts as a shield as you're holding the left trigger, which is nice, but far from essential.
Vah Medoh gives you the ability to create powerful updrafts to scale peaks with ease, which is immensely useful when exploring the world. Vah Naboris gives you the power to summon powerful lightning strikes against all enemies in a fairly large radius around you. That one almost breaks the game with its ability to make fights against giants, Lynels and other boss-type enemies much easier. After Vah Ruta, I recommend going after whichever one of those sounds more appealing to you. They're both life savers.
More Zelda: Breath of the Wild tips, tricks and guides
Find out all there is to know about Zelda: Breath of the Wild, including what to expect from the Wii U version, how to preserve your items, how to beat bosses like the Stone Talus, Lynel, the Guardians and the Hinox, the best recipes for Link and how to take on the game's shrines. You'll also want to find out where all the great fairies and available stables are in the game, how to use amiibo with your version of Zelda and how to increase your health, stamina and weapon slots.