Is there a full list of post-game "activities" or events I can participate in after becoming the Island's Champion?
Not currently aware of a full list, compiling one below. Possible spoilers and bad jokes.
Collect all the Cells
Zygarde collection quest. You can finish collecting the Zygarde Cores/Cells, some of which are only accessible after becoming champion.
Help the International Police
Lookers Quest. Looker needs help dealing with the Timey-Wimey Ultra Beasts, and Hau's not going to do it.
Gather Island Guardians
Catch the Tapus. Its not like they would be saving people if they weren't trapped in your PC box.
Explore Poni Island
Battle on a tree. Face off against trainers including the legendary Red and Butts, I mean Blue. Get a 100 win streak to win a ... berry? Or get the full 200 win streak to get a different berry.
Battle Morimoto. Once per day you can take on his arbitrary selection of Generation 1 Pokemon.
Complete the Pokedex
Arguably only completable in postgame due to legendaries. If you're not completionist, you won't be able to get the Shiny Charm for that sweet 0.073% shiny encounter rate.
Compete in Battle Royal. Could be done before the beating the League but you didn't do that. Rage at the AI for stealing your win right before the end of the match. Win totally functional ribbons for completing Master rank.
Mighty Morphing Mega Pokemon. You get the Keystone for mega evolution, as well as Alakazite after beating Dexio on the Ancient Poni Path. Now try to find all the Mega Stones! Try at the Battle Tree, and yes you have to buy them.
Take on New Challengers
When beating the Elite Four in the Post-Game, you will face one of 10 random Challengers as your fifth match. They include Hau, Sophocles, Ryuki, Gladion, Molayne, Plumeria, Hapu, Faba, Tristan, and Kukui.
Other Post-Game Trainers
Definitely incomplete. Some trainers can be fought post-game with high level Pokemon, and they may have a reward for beating them.
Beat up the Trainer School principal on Route 1 to get a King's Rock