While you have some spare time, there are a few new things you can do around Akala Island, which are easily missed.
Match with Lana
Curious about how strong the captains of Akala Island? Now's your chance to find out! To battle Lana, head for her house in Konikoni City, north-west from the Pokemon Center. Inside, enter the room at the far end. You'll find a pair of twins who want to battle you, no matter what.
Twins Harper and Sarah
A Double Battle with two Water-types. Show them who's boss with your Grass or Electric-types.
Once you're done playing around, Lana will arrive. Needless to say, you'll be dragged into a battle.
As you'd expect, all of Lana's Pokemon are Water-types, but they're not easy Water-types.
Chinchou is part Electric and has Volt Absorb so keep your Electric-types away. Grass-types are best, while Pokemon that aren't weak to Electric or Water ought to survive.
Shellder is pure Water, but has Icicle Spear and Skill Link, which can result in much pain for Grass-types. Electric-types are the way to go.
Finally, Araquanid is part Bug and thus weak to Electric, Rock and Flying Pokemon. Of course, Rock Pokemon should be benched since they're weak to Water. It also has Waterium Z and boosted Water damage due to Water Bubble.
Sink Lana's ship and you'll receive a Muscle Band for your time. You will also get the inspiration for the "Girly" battle style. Which may not be useful now, but will be later.
Match with Kiawe (Sun)
If playing Pokemon Sun, make your way to Paniola Town during the night. Once there, head inside the building west from the Pokemon Centre. Upstairs, go inside the top-most room and you should find Kiawe practicing his fire dancing.
After a brief chat, he will leave for Thrifty Megamart. Exit the building and use Charizard Glide to instantly reach Royal Avenue. From there, travel north-east towards the Megamart. You will find Kiawe by the north-east corner. Approach to eavesdrop.
All Pokemon you'll have met before, hopefully. All Fire-types too, naturally.
Growlithe is the easiest here, target practice for any of your Water, Rock or Ground Pokemon.
Fletchinder is a bit stronger, but can't hold a torch to Rock-types. Electric and Water work well too.
Finally, Marowak is weak to Water, Ground, Rock, Dark and Ghost. You've got plenty of choices, so don't sweat it. Physical attacks need to beware of Will-O-Wisp though.
Afterwards, Kiawe will leave on his way. You may not notice at first, but you'll get the inspiration for the "Reverent" battle style.
Match with Mallow (Moon)
This one's only available for players of Pokemon Moon. When it's day time (which is night time in Moon), head for the restaurant in Konikoni City, south from Olivia's store. Go upstairs and you should find Mallow in her room. Offer to play and she'll run ahead to the Lush Jungle.
Follow her there and chat to the Wimpod up ahead. Mallow will arrive on the scene and demand a battle.
All Grass-types obviously. Having Fire, Bug and Flying Pokemon will be instrumental to success.
Phantump is weak to Fire, Flying, Dark and Ghost Pokemon. That should be plenty of options and it's pretty weak so you shouldn't have a hard time.
Shiinotic is susceptible to Fire, Flying and Steel Pokemon and takes 4x damage from Poison-type moves.
Steenee is a pure Grass Pokemon, so you don't really need a lesson on that. It's also a middle evolved form, so not particularly strong either.
Afterwards, Mallow will leave in a hurry. While it may not mean anything right now, you'll get the inspiration for the "Reverent" battle style.
Meal with Olivia
This is pretty minor, but cute and easy to miss. Head to Konikoni City and visit the restaurant south from Olivia's store. Speak to the woman by the counter and order the "special Z-set" for 1,040 Poke Dollars.
After placing the order, Olivia will come in. Since you're both here, she decides to dine with you. At the end, you'll receive 4 Heart Scales, instead of the usual two. If you'd like, you can repeat this event once a day, making it a free and easy way to farm Heart Scales.
Remember the Police Station on Route 9, by the south entrance of Diglett's Tunnel? Head inside and chat to the Snubbull by the entrance. This will alert the Police Officer on break. Before, he shooed you off, but for some reason he's a lot more focused.
Chat to him to receive a Focus Band. Give this to a Pokemon and it may survive a KO with 1 HP. It's not terribly reliable, but who knows, it might come in useful.
Gift from Burnet
Another minor thing, although potentially more important in the long run. Travel to Heahea City and pay another visit to the Dimensional Research Lab, west from Hano Grand Resort.
Upstairs, if you chat to Professor Burnet in her office, you will receive the Reveal Glass. This key item lets you change the form of the legendary Pokemon Landorus, Thundurus and Tornadus. Fitting since Burnet was researching them in Dream Radar.