Having 317 species of shinies in my Dex and over 350 in my boxes, I feel inclined to answer this.
Shiny hunting is arguably the easiest it's ever been in the main series games, and the chances you'll find one in a normal playthrough is higher than in past titles. Of course, there are methods you can use to increase your odds of finding one of these elusive differently colored creatures, so without further ado, here are:
The Methods
Getting the Shiny Charm
We're all familiar with this one. Bringing your base shiny odds from 1/4096 to 1/1366, this is definitely something you'll want as it boosts the odds of other shiny hunting methods, which will be mentioned later. Requiring 400 Pokemon in the dex, the Shiny Charm isn't too hard to get, especially with the inclusion of Tera Raids, but of course, there are still version exclusives to trade for and a couple mons that may be tricky to get. Just talk to Jacq (your biology teacher) in the academy after completing the Pokedex to obtain the Shiny Charm.
The Masuda Method
Another familiar method. Breed 2 Pokemon of different languages and the odds of you hatching a shiny fom an egg goes up to 1/683, or 1/512 with the Shiny Charm. You'll typically want a foreign Ditto, as it can breed with most Pokemon in the game. A Pokemon with Flame Body (like Talonflame) is also useful for speeding the process of hatching eggs.
In this game, breeding works a bit differently. Set up a picnic with 2 Pokemon that can breed (you can have more if you want to do a multi-hunt, but generally, you'll want just 2 if you're going for something specific), make a sandwich with Egg Power (I usually use sandwich 110 to get Egg Power 2) to boost the chance you'll receive an egg, and occasionally check the basket at the end of the table for eggs. The basket can store up to 10 eggs at a time, so make sure you don't forget to check it. Make sure you recall your Koraidon/Miraidon or Ditto won't breed with your other Pokemon. Additionally, don't forget to save after you the sandwich eggs, in case one of the eggs happens to be shiny.
Sparkling Power
A new method to increase shiny odds, making a sandwich with 2 Herba Mysticas, and you'll get Sparkling Power 3, which boosts the odds of finding a shiny to 1/1024, or 1/683 with Charm. When you make a Sparkling Power sandwich, the timer will pause if you put your console in sleep mode, but if you close out of the game, the sandwich boosts will be gone. The Herba Mystica odds also stack with Outbreaks, which will be discussed soon.
Herba Mysticas are obtained by defeating 5 and 6 star raids. However, they can only be obtained post-game, and it isn't guaranteed you'll get them. 5 star raids have about a 10% chance to drop Herba Mysticas, while 6 star raids have about a 13% chance. Making a sandwich that gives you Raid Power does give you a chance to get more Herba, though.
Mass Outbreaks
A fairly newer method. Head to an outbreak and knockout 30-59 Pokemon to boosts the odds to 1/2048, 1/1024 with Charm, 1/820 with Sparkling Power 3, and 1/586 with Charm and Sparkling Power 3. Knockout 60+ to bring your odds to 1/1366, 1/820 with Charm, 1/683 with Sparkling Power 3, and 1/512 with Charm and Sparkling Power 3. Using the autobattle feature is the easiest way to quickly KO Pokemon. When you get the message that there aren't many Pokemon left from the original outbreak, that's usually a good indicator that you've defeated more than 60 Pokemon from the outbreak, however I do recommend keeping a mental count. Be careful not to defeat too many Pokemon or the outbreak will end.
Tips, Tricks, and Knowledge
Keep an eye out for shinies
If you're someone who cares about not missing a shiny, then keep an eye out for them, because since they spawn in the overworld with no indicator that they're nearby (sparkles or a noise), you could run right past one without knowing. Use the autobattle feature to seek out hard to spot ones like Igglybuff, Tandemaus, Flabébé, and others, because the autobattle feature won't kill shinies. Look in the trees because some Pokemon will spawn in them. Additionally, Pokemon found in Tera Raids (excluding event ones) can be shiny, but they won't sparkle, so catch harder to notice ones like Blissey and Gengar just in case. You can invite friends to Tera Raids with shinies, but if you aren't connected to the internet before when you find a shiny raid, and your event raids aren't up to date, do NOT connect to the internet because your raids will reset. Don't turn your camera recklessly, because Pokemon tend to spawn with the camera, and you don't want to be running towards the camera, because a shiny could spawn in the background and despawn instantly.
Research what you're hunting and properly prepare
Some shinies can be hard to spot if you don't know what you're looking for, so look at a picture online so you don't miss it. Additionally, look at a Pokemon's learnset, because some Pokemon carry moves like Self-Destruct, Memento, and Final Gambit, which can lead you to failing it. Get a Pokemon with Damp for Self Destructers, a Ghost Type for Final Gambit (along with Normal Type recoil moves like Double Edge), and Taunt for Memento. Notable users of these moves are the Drifloon, Voltorb, and Glimmet lines (Self-Destruct), Hoppip and Stunky lines, Tatsugiri (Memento), and Veluza (Final Gambit).
Zorua, Zoroark, and Ditto disguise themselves as other Pokemon in the overworld. There are some indicators (a Zorua disguised as Murkrow won't have a green question mark above it's head when you go near it and will run away, for example), but you're definitely going to want to use autobattle to identify the posers.
SAVE!!!
When you find a shiny, save the game so you can reset if you fail it. Sometimes it won't appear right away when you reload the game, so you may have to reload the game several times. Before you make a Sparkling Power sandwich, save so that if you don't find a shiny you can reload the game. You never know when something could go wrong, so save just in case.
Check your surroundings
Scarlet and Violet aren't perfect games, and Pokemon can spawn in the most obscure places. Make sure you check in the walls and even in some rocks (I found a shiny Meowth in a rock) by holding ZL and tilting your camera, because Pokemon can and will spawn in strange spots. Additionally, make sure your outbreaks spawn a decent amount of Pokemon, because some outbreaks just aren't good spots.
Know how to use the Herba Mystica
Depending on the ingredients you use, you can make a sandwich that boosts the Encounter Power for a specific type, which is perfect for hunting a specific Pokemon in a specific spot. There are many different recipes, so make sure you look online to find one for the type you need.
Edit: As of update 2.0.1, the way outbreaks work is different. Outbreaks no longer spawn based on your location, meaning you can't manipulate your outbreaks to look for a specific Pokemon (EX: resetting in the Socarrat Trail area with a grass encounter sandwich to try to get a hard to find outbreak like Leafeon). However, encounter power sandwiches do still affect outbreaks, meaning you can get a bunch of specific-type Pokemon outbreaks all throughout the map, which is good for going after multiple targets consecutively, but not nearly as effective for trying to get a specific mon in an outbreak.
Despawning Pokemon
Setting up a picnic despawns everything, so using that can make encounters go by a lot quicker. The same thing applies with going into a town. Just be careful not to despawn a shiny.
Prepare to catch Pokemon
Just use basic preparation. Bring a False Swiper, Status Moves, a bunch of Poke Balls, that kind of stuff (or you could just spam Poke Balls and hope for a crit capture, but that's not reliable).
And with that, I think I covered everything. Good luck on your hunts!
