After defeating all the Island Trials on a given Island, you must first defeat that Island's Grand Trial before moving on to the next Island full of even more Island Trials. Each Grand Trial is nothing more than a battle, but with the Kahuna of each given Island. These trainers are among the best in all of The Alola Region, so be prepared. After defeating all four Grand Trials, you will be permitted to challenge the Pokemon League.
Grand Trials
After making it back to Iki Town - having completed Captain Ilima's Trial and helping Lillie - you will face Kahuna Hala in your Melemele Island Grand Trail. Hala is an expert Fighting Type Pokemon User, having the power of Z-Crystals as well. Defeating him will earn you a stamp on your Trainer Passport, forcing traded Pokemon up to Level 35 to obey you. You will also receive Fightingium Z for defeating Hala, as well as the Rider Pager you can use to ride Tauros.
Island Kahuna Hala - Level 15 Machop, Level 15 Makuhita, Level 16 Crabrawler
After defeating Plumeria and getting to the end of Akala Outskirts, you will arrive at the Ruins of Life. Directly outside of them is where you will battle Kahuna Olivia, the woman you met right after arriving on Akala Island. Lillie and Burnet will arrive before the battle begins, with Lillie actually watching the battle itself. Olivia is a Rock Type Pokemon User, one that can use the power of Z-Crystals
Kahuna Olivia - Level 27 Anorith, Level 27 Lileep, Level 28 Lycanroc
After defeating her, you will earn the next Stamp on your Passport that will cause all Pokemon up to Level 50 to obey you. Olivia will also give you Rockium Z.
As you meet Gladion on the ferry dock in Malie City in hopes of finding Lillie, Nanu will appear and challenge you real quick to a battle. It turns out he's actually the Kahuna of Ula'ula Island, and this, no matter how rushed and unexpected, is your Grand Trial. He is a Master of Dark Type Pokemon, one with access to a Z Crystal.
Kahuna Nanu - Level 43 Sableye, Level 43 Krokorok, Level 44 Persian
After defeating him, you will receive your next Passport Stamp that will cause all Pokemon up to Level 65 to obey you. Kahuna Nanu will then give you the Darkinium Z-Crystal.
After completing Captain Mina's Trial, you can head back to Exeggutor Island to have your Poni Island Grand Trial with Kahuna Hapu. If you didn't already know this, Kahuna Hapu is a Ground Type Pokemon User and has access to Z Crystals. Defeat her to receive your next stamp of your passport - allowing Pokemon up to Level 80 to obey you - as well as Groundium Z.
Kahuna Hapu - Level 53 Golurk, Level 53 Gastrodon, Level 53 Flygon, Level 54 Mudsdale