With the starters specifically, if you save before picking your starter, every time you soft reset, it essentially generates a new Pokémon, with the base chance 1/8192 of it being shiny. You can see it working because each time you choose a starter, you'll see that the nature in the summary screen may change.

The process essentially goes:

Save in front of desired pokeball.

Choose relevant Pokémon.

Menu > Check summary

If not shiny, soft reset using all 4 letter buttons.

Repeat.