What increases your chances of finding shiny pokemon?
The Shiny Charm returns once again this Generation and increases the regular encounter shiny rate from 1 in 4096 to 1 in 1365.3. It can be obtained by completing the Pokedex and then talking to a doctor within a room on the 2nd floor of Hotel Ionia in Circhester. The Masuda method has also returned with no significant changes as far as we know. Chaining however has been replaced by a new system:
The more times you've defeated a certain species of Pokemon in battle, the higher your chances are of finding a shiny of that species, with the best odds being at 500+ battles. These battles do not have to be in a row (chained) to receive these increased odds. You can keep track of how many times you've battled a certain species of Pokemon from their Pokedex entries as it has a counter. For example, at 500+ battles won against one species, your chances of getting a shiny are increased to 1 in 683 (1 in 512 with the Shiny Charm). The graph at the bottom of this page should help. *
*Serebii, which has been a reliable source of Pokemon information for years, is claiming it's possible there's a bug in the code causing the new shiny hunting method not to be anywhere near as effective as it should be. In the table linked you linked there is a "Chance of Shiny Rate" column. This column shows that at 500+ encounters you only have a 3% chance of receiving the boosted 1 in 683 odds. Serebii has also confirmed that's what the column means on their official Reddit account. Just some anecdotal data from me, I reached 6000 encounters of the same Pokemon without finding a shiny. And from YouTubers I watch they seem to be having much more luck with the Masuda method. So I'd stick to Masuda for now if I were you.
You can increase your chances of finding shiny pokemon during breeding and battling.
Regardless of which method (or both) you're using, you would want to get the Shiny Charm.
So first you'll have to complete the Pokedex to receive the Shiny Charm from a doctor on the 2nd floor of Hotel Ionia in Circhester. This will triple your base shiny chances from 1/4096 to 1/1365.33.
Breeding
Use Masuda method by getting a shiny Ditto of a different language. Country does not matter.
Also, as chances are determined when you receive the egg, it is more efficient to hatch it with a hot-bodied pokemon than it is to do the save-hatch-restart method.
Battling
There's no chaining for shiny pokemon in this version. Even the link in the other answer doesn't make any mention of it.
It's based on battling, and it maxes out after knocking out/catching the specific pokemon in 500 battles, which gives 6 times the base chance, i.e. 1/682.6. Combine this with the Shiny Charm for maximum results - 1/512 chance.
Source: https://www.pokemon.com/uk/strategy/gigantamax-shiny-pokemon-sword-pokemon-shield-wild-area/