As far as I know, it doesn't really continue. There are side quests you can do in the meantime, along with filling out the pokedex, training up your Pokemon, and battling in the battle tree. There is this side quest I discovered in the Thrifty Megamart on Royal Avenue. If you talk to the Veteranat the first counter (or last, depending on which entrance you come through), he tells you about how when he was younger he tried to defeat eight trainers who each used a different evolved form of eevee. Then he gives you hints on where to find each of them. There's probably many more side quests you can do like that one to pass the time. Hope this helped :)
