After going through Diglett's Tunnel and passing through Route 9, you go to Konikoni City to meet Olivia.
In the Konikoni Pokémon Center, there is a guy from the Aether Foundation who asks you to catch a Passimian (in Sun) or Oranguru (in Moon), which can be found in Lush Jungle. If you show him your Pokédex afterward, he will pay you 5000 Pokédollars.
In the Pokémon Center, the little girl, Laule'a, asks you to trade your Zubat for her Poliwhirl, Whirly.
Olivia's shop is across from the Pokémon Center. When you go inside, Olivia's Probopass gives you a letter. Olivia asks you to go through Memorial Hill to meet her at the Ruins of Life. Then Probopass gives you a Max Potion.
At Olivia's shop you can get fossils, evolution stones, and accessories. The woman selling the fossils makes it sound like you can only choose one, but you can actually get both—you just can't get more than one of the same type. Each one costs 7000 Pokédollars. You can take the fossils to the Fossil Restoration Center on Route 8 to have them turned into pokémon.
Go upstairs in Olivia's house and go into the room there to find a pink sparkle for your Zygarde Cube.
There is a green sparkle for your Zygarde Cube at Lighthouse Point to the north.
Talk to the woman at Lighthouse Point to get Pikanium Z.
To the right of that woman there is Eviolite.
After you are done exploring, go southeast to Route 9, and southeast from there to reach Memorial Hill.