Your Akala Island adventure starts in Heahea City. After your welcoming, you're free to wander around, somewhat. Before you can move though, you buddy Rotom lets you know that your Pokedex has been expanded to include the Akala Island Pokedex.
From the docks, the first building on the right is Akala Island's Ferry Terminal. Inside, you can speak to the clerk at the back to book a ferry back to the Melemele Island Marina. Also, if you check the vending machine in the top-right corner, you can buy Fresh Water.
Further right is the Tourist Bureau. If you regret any of the nicknames you've given to a Pokemon, you can speak to the Name Rater next to the face board to give it a new name. Keep in mind that you can't change nicknames of Pokemon from trades.
Keep following the road and you'll soon bump into Lillie in front of the apparel shop. The two of you will chat and then Lillie will go on ahead to the Tide Song Hotel, north from the apparel shop. Once you're back in control, you can check out the apparel shop if you want.
Inside, there's a man in the bottom-right corner who really wants to see a rare Pyukumuku. You can catch one in the beaches of Hano Great Resort, much later on. If you do that and remember to come back to report to the man, you'll earn 10,000 Poke Dollars.
Opposite the apparel shop, be sure to speak to the woman standing by the edge to receive a free Rare Candy. Next, carry on east towards the Pokemon Centre. If you try to head further east from here, a trainer riding a Stoutland will be blocking the way.
Inside the Pokemon Centre, you can purchase protective TMs from the right clerk of the Poke Mart. Also, if you didn't notice earlier, the standard (left) clerk will be selling more items now that you've cleared Ilima's trial. Stock up if you must, because you've got a long journey ahead!
Back outside, follow the road between the apparel shop and the Pokemon Centre north. Just before you reach a T-junction on the right, a pair of trainers will catch your attention. Why, it's Dexio and Sina from Pokemon X and Y. But what are they doing here? Testing you of course!
Pokemon Trainer Dexio (Sun)
Slowpoke isn't an immediate threat and can be swiftly taken out with Grass, Electric, Bug, Ghost or Dark-type Pokemon. Espeon is no problem for a Dark-type. Bug and Ghost types will be good as well, as long as they're not Poison as well.
Pokemon Trainer Sina (Moon)
Delibird is pretty much a joke, but to be safe, have your Fire, Electric, Steel or Rock-types take it out. Glaceon is a bit tougher, but Fire, Fighting, Steel or Rock Pokemon are more than enough to stop it in its tracks.
Pass the test and you'll receive an enigmatic Zygarde Cube. This key item allows you to collect the 5 Zygarde Cores and 95 Zygarde Cells scattered across Alola. Later on, you can use the Cores and Cells to build a legendary Pokemon known as... duh, Zygarde.
The reason we suggested waiting to re-explore Melemele Island is because a dozen Zygarde Cells can already be found there. Of course, you weren't able to collect--or even see--them until now. Anyway, it's no mean feat collecting all 100 Cores and Cells, but we've got your back!
For now, continue ahead, past the water fountain, to reach Tide Song Hotel where Lillie is waiting. Inside, head all the way to the top-left corner of the hotel. Above the dancer lady, you should notice a sparkling green object in the corner. Inspect it to find Zygarde Cell (15).
More in Melemele
Wait, Zygarde Cell... number 15? Yup, that's because there are 14 other Zygarde Cells found in Melemele Island, although not all of them are currently accessible. In fact, one of them is only available after completing the game!
Anyway... for now, head to the other side of the hotel, to the right of the main reception. You should find Lillie looking anxiously around. After chatting to her, exit the hotel. From here, you can head east from the T-junction below the water fountain to reach Route 4.
Or... you can return to Melemele and go looking for some of those elusive Zygarde Cells, in addition to exploring some areas you may not have visited. If so, flip back to Melemele Sidequests.