This research line unlocks after reaching Step 6 in the A Mythical Discovery Research Line.
Evolving an Eevee to an Espeon/Umbreon for steps 4 and 5 respectively can be a bit annoying due to the walking requirement for them, and those who haven't done so already may be tempted to utilize the Nickname Trick to get them quickly (Check Eevee's Page for details), but the quest also requires that players walk 10 km to earn a candy with their Eevee buddy, so it's generally best to just work through it.
Visiting Pokestops 7 days in a row for the 6th step puts a time-gate on this Research Line, and can also be a bit infuriating if you happen to miss a day. For those who don't pass by Pokestops on their daily commutes, it might be best to set a reminder to go out and spin a stop every day.
The other bottleneck can be the Gold Johto Metal requirement on Step 7. Johto has many common Pokemon that show up all around, but it also has a few rare Pokemon that can make getting the gold medal a bit difficult. Your best bet is to keep an eye on the current/upcoming events to make sure that you can target any currently uncaught Johto Pokemon, as well as gather candy to evolve whatever may remain. Fortunatly, this line also offers a lot of evolution items relevant to the Pokemon from Johto, so that should help remove some RNG from the proceedings.