Once you've cleared Nanu's Grand Trial, a few new things will occur on Ula'ula Island. Not a lot, mind you.
Mimikyus Unwanted Clothes
In the Malie City Pokemon Centre, chat to the girl with the Mimikyu to receive a Reaper Cloth. This sinister item will evolve Dusclops into Dusknoir if it's traded while holding one. You won't find either in Alola though; they'll need to be gotten from another game.
Love and Hate
When it's night time, go inside the malasada shop in Malie City. In the corner next to the checkout is an Oranguru. Chat to him to receive TM21 Frustration. Afterwards, his owner will come along and give you TM27 Return as well.
Frustration and Return are both Normal-type moves that deal more damage the more your Pokemon dislikes or likes you respectively. Frustration is best used for new Pokemon with naturally low friendship, while Return is best used with your long-time companions.
Meal with Nanu
This is similar to the event in Konikoni City. If you visit the Sushi High Roller in Malie City and choose the fourth option, the "Z-Kaiseki: Ronin Set", Nanu will come along and dine with you. At the end, you'll receive 8 Heart Scales instead of the usual 4.
Like the Olivia event, you can repeat this particular event once a day. Which you should totally do if you remember, since you get a whole bunch of Heart Scales for next to no effort. Unless, of course, you're swimming in Heart Scales; in which case, we don't blame you.
Remember the kid in the Malie Community Centre flea market with the Shiny Stone (Sun) or Dusk Stone (Moon)? Visit him again and he'll finally decide to sell his prized possession for 3,000 Poke Dollars.
The Shiny Stone can be used to evolve the following Pokemon: Togetic, Roselia, Minccino and Floette. None of these Pokemon are native to Alola, although Roselia and Togekiss (evolution of Togekiss) can be found using the Island Scan.
Meanwhile the Dusk Stone evolves Murkrow, Misdreavus, Lampert and Doublade. The former two can be found in Alola, while the remaining two will require the Island Scan.
Secret of the Pond
When it's day time, make your way to the pagoda in the north-west corner of Malie Garden. It's the one in the lake between the bridge and the narrow, winding path. Chat to the Lass standing by the edge and answer "Gyarados" to receive an Air Balloon.