No raids or DLC
Look for an Eevee outbreak. When playing for the first time each real-life day, check the outbreaks. Restart the game without saving if you do not see Eevee (or Sylveon). Should see new outbreaks upon loading back in. May take a while to find Eevee, however, even with this method.
In the event of seeing an Eevee outbreak: save. Grab a high level Pokemon and head over to the outbreak. (Personally likes Chien-Pao for this. Acts aggressively. Manages fine with just about any Pokemon, though.) Send out your Pokemon with the Let's Go feature and faint Eevees. (Do not worry about killing a shiny. Will not do that in Let's Go mode.) Keep going until you see a message that says "There are not many Eevee left from the original outbreak". Avoid fainting any Eevees from this point onwards.
Look around. Do you see a shiny Eevee? If no, start a picnic and immediately end it. Clears all the Pokemon around you and spawns new ones. Continue until you find a shiny. Be careful
not to start a picnic if a shiny is there. Will despawn it, just like any other Eevee.
Supposedly finds a shiny at a 3/4096 chance (5/4096 with the Shiny Charm) with this method.
Without raids, but with DLC
Head to the Blueberry Academy cafeteria. Buy the Academy Special for 150 BP. Yields Sparkling Power Level 1. Save before buying it, also. Allows you to reset if you do not find the shiny Eevee before the power expires. Spends less time doing Blueberry quests that way.
Recommends looking for an Eevee outbreak first with this method.
With raids and Herba Mystica
Do you have any Pokemon at level 100? If not, try doing some 4-star raids with Koraidon/Miraidon. Recommends grabbing the one from the bottom of Area Zero, rather than the one you traveled with. Hopefully allows you to build up some EXP Candies to tackle some 5-star raids.
- For Koraidon: Drain Punch, Swords Dance (TM move), Collision Course, and Dragon Claw. Prioritize raids that are weak to Fighting. Give it a Shell Bell (from Levincia's Delibird Presents) or a Big Root (near a big tree near Cortondo, apparently).
General strategy during raids: Lead with Swords Dance 3 times. Maximizes your Attack. Use Drain Punch in-between if you ever need a heal. Terastallize if you are not close to dying and launch a bunch of Collision Courses. May need to use Swords Dance again (ideally 3 times) if the raid boss clears your stats.
- For Miraidon: Parabolic Charge, Electro Drift, Metal Sound, and Calm Mind (TM move). Focus on raids weak to Electric. Give it a Shell Bell (from Levincia's Delibird Presents) or a Big Root (near a big tree near Cortondo, apparently).
General strategy during raids: Hit the opposing Pokemon with Metal Sound while the Tera Shield is not up (3 times at most). Consider a few Calm Minds if the Tera Shield is up and you are not dealing big damage (6 times at most). Fire off a Parabolic Charge if you ever need healing. Terastallize when you can if you are not about to faint. Blast it with Electro Drifts from there.
For all Pokemon: Never use any cheer besides Heal Cheer. Typically reserves a Heal Cheer for a very immediate heal OR to remove status like Sleep.
Do you have lots of money? Purchase vitamins at the Chansey Supply in Levincia (and several other locations).
- Feed Koraidon up to 26 Protein and 26 HP Up.
- Feed Miraidon up to 26 Calciums and 26 HP Up.
Offers advice for any other high-level Pokemon you may have too.