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 Island Trials. Each Grand Trial is nothing more than a Battle, but with the Kahuna of each given Island. They are among the best trainers in all of Alola.
Grand Trials
After completing Captain Ilima's Trial, once you return to Iki Town, Hala will challenge you to your Grand Trial. He is a Master of Fighting Type Pokemon, having a Level 14 Mankey, Level 14 Makuhita and a Level 15 Crabrawler with Fightinium Z. Defeat him to earn the right of all Pokemon up to Level 35 to obey, the Fightinium Z-Crystal, and the Rider Pager. You will also get the next stamp of you passport.
After completing the three Island Trials on Akala Island, Lillie will ask you to take her to the Ruins of Hope. Once there, Olivia will challenge you to your Grand Trial. She is a Master of Rock Type Pokemon, having a Level 26 Nosepass, Level 26 Boldore, and a Level 27 Lycanroc with Rockium Z. Defeat her to earn the right of all Pokemon up to Level 50 to obey, the Rockium Z-Crystal, and your next passport stamp.
After completing the two Island Trials on Ula'ula Island, right before you leave the island with Gladion, Nanu, the Kahuna of Ula'ula Island, will challenge you to your Grand Trial. He is a Master of Dark Type Pokemon, having a Level 38 Sableye, Level 38 Krokorok, and a Level 39 Alolan Persian with Darkinium Z. Defeat him to earn the right of all Pokemon up to Level 65 to obey, the Darkinium Z-Crystal, and the next stamp on your passport.
After arriving on Poni Island, you will quickly find that there is no Kahuna on Poni. The previous one was Hapu's Grandfather, who recently passed away, and Tapu Fini has yet to pick another one. However, after going with Hapu to the Ruins of Hope to request for another Kahuna to be chosen, Hapu herself will become the New Kahuna. She will finally Battle you in front of Vast Poni Canyon.
She is a Master of Ground Type Pokemon, having a Level 47 Dugtrio, Level 47 Gastrodon, Level 47 Flygon, and a Level 48 Mudsdale with Groundium Z. Defeat her to earn the right of all Pokemon up to Level 80 to obey, the Groundium Z-Crystal, and the next stamp on your passport. Defeating her will also complete your Island Challenge, which ultimately gives you the right to challenge the newly created Elite Four.