There are various ways to get each stone, but these should be pretty accessible while going through the main story.
Fire Stone
Method 1: Buy in Konikoni City in Olivia's store
Method 2: Find in Diglett's Tunnel on Akala Island
Ice Stone
Method: Po Town, top left
Leaf Stone
Method: Buy in Konikoni City in Olivia's store
Thunder Stone
Method 1: Dream Park on Akala Island, behind the truck
Method 2: Buy in Konikoni City in Olivia's store
Water Stone
Method: Buy in Konikoni City in Olivia's store
Dusk Stone
Method 1: Flea Market in Malie Community Center
Method 2: Poni Wilds beach on Poni Island, hidden underwater
Moon Stone
Method: Route 13 on Ula'ula Island, by the fountain
Sun Stone
Method: Ula'ula Island, Blush Mountain
Oval Stone
Method: Paniola Ranch on Akala Island, near the Pokemon Nursery
Dawn Stone
Method: When in the post-game, talk to Guzma in his house (it's south of the berry farms), then after following his directions and beating him in a battle, he'll give you a Dawn Stone
Also: any of these stones can be found by using Poke Pelago's Isle Aphun (at least level 2) and selecting the Brilliant Stone option to explore.
Source
Source
