Lillie drops by your house the next day because Professor Kukui has asked her to escort you to his Lab. It's not far; just head south from your house to find the Lab near the beach. Before you go, your mom gives you some pocket money for your journey.
Feel free to catch more wild Pokémon in the tall grass that you can trudge through along the way—new species await discovery here.
You have no means of traveling across water currently, so you can't catch Water-type Pokémon (minus Wingull and Slowpoke) or find every last item in this area just yet. The time will come when you can, but for now, simply explore the accessible areas as you continue on our journey.
On your way to Kukui's Lab, you'll run into three trainers.
Professor Kukui is hard at work inside his Lab, but he pauses to introduce you to Rotom, a Pokémon that lives inside of your Pokédex. Kukui then upgrades your Pokédex, enabling Rotom to talk to you and guide you. This upgraded Rotom Pokédex will help you stay on course as you explore the Alola region.
The Trainers' School is your next destination. When you're ready to move on, leave the Lab and go north towards your house.
A cutscene will happen when a solarDay/lunarNight eclipse suddenly occurs. Afterwards, head west along the road to reach the Pokémon Center. Lillie is waiting out front for you.
Talk to the nurse at the main desk and she will heal your Pokémon for you. The Poké Mart sells a variety of items to help you on your adventure.
The café owner will give you a Rare Candy for each milestone of total Poké Beans fed to all of your Pokémon. The first time you buy a drink from any Pokémon Center Café each day, the owner will also give you an item and 12 Plain Beans, depending on the day. Each drink costs 198.
Also, once you have obtained more than six Pokémon, you'll have to send the extras to your PC boxes. You can rearrange your Box Pokémon and your party however you'd like.
Now that you've received the Rotom Dex, new options have appeared on your X menu:
Quick Link, which allows you to immediately battle friends or trade with them.
Festival Plaza, which is full of shops and activities where you can communicate with trainers around the world and earn Festival Coins.
Battle Video, where you can record your battles you've participated in via Quick Link or Festival Plaza.
QR Scanner, which lets you easily register Pokémon in your Pokédex by scanning QR Codes.
Festival Plaza[]
Festival Plaza
Sophocles will ask you to help with the festival. Someone will arrive in the plaza. Talk to this person and you get two Festival Coins.
Talk to Sophocles again and he will show you the facilities in your Festival Plaza. This can include places like a lottery shop, a bouncy house, a fortune-teller, a haunted house, a dye house, and more. More facilities will open up as you play and meet more people in your Festival Plaza.
Two more people arrive in the plaza, so talk to them and respond to their questions. Afterward, your plaza will go up to Rank 2. Talk to Sophocles and a new facility will be added to your Festival Plaza.
Go into the castle to meet the staff there. You can get information from the receptionist, participate in Global Missions, page specific visitors, and rearrange the facilities.
Right next to the Pokémon Center is the Trainers' School. An ornery Tauros blocks the road ahead, but the Trainers' School is still accessible. Go there to find Professor Kukui waiting for you. He challenges you to defeat four trainers. He then gives you the special item—the Exp. Share. With it, EXP points will automatically be distributed to all party Pokémon from now on. The Exp. Share can be turned on and off at will. It is turned on by default.
There are four trainers scattered across the school grounds that you need to battle. Lillie will heal your Pokémon if you need it. In the bottom right of the building is a Potion. The first trainer stands northeast from there.
Southwest from her is another Potion. At the battle arena, the boy won't battle you until you defeat the other trainers. Go northwest and into the red gate. Southwest of the gate is an Antidote. Go south of that to find a Paralyze Heal. The next trainer stands just south of here.
After you've beaten the two trainers around the grounds, enter the school and talk to the girl in white to get a Quick Claw. Go left all the way and into the classroom. The little Preschooler girl in front of the classroom on the left will battle you.
After defeating him, he'll reward you with your first TM: TM01 (Work Up).
After defeating them all, you're suddenly called to the school's office. Go inside and upstairs. Go left and into the second classroom. One of the students gives you a Potion. Go to the stairs to find Teacher Emily waiting for you. She's eager to learn how you managed to defeat her four top students and challenges you to one more battle.
After defeating her, she gives you five Great Balls. Kukui arrives after your battle and introduces you to Ilima, the first Trial Captain you will encounter in Alola. He runs his trial in Verdant Cavern. Ilima looks forward to your challenge. For now, follow Lillie into Hau'oli City.
You'll encounter Tauros before entering Hau'oli City. Help Hala get the ornery Tauros out of the way. Choose to pet its face and it'll move, allowing you to get to the Hau'oli City Beachfront.
Awakening. Go south to battle a Lass.
Rotom Pokédex will help you stay on course as you explore the Alola region.
Rare Candy for each milestone of total Poké Beans fed to all of your Pokémon. The first time you buy a drink from any Pokémon Center Café each day, the owner will also give you an item and 12 Plain Beans, depending on the day. Each drink costs 
198.
Exp. Share. With it, EXP points will automatically be distributed to all party Pokémon from now on. The Exp. Share can be turned on and off at will. It is turned on by default.
Potion. The first trainer stands northeast from there.
Antidote. Go south of that to find a 
Paralyze Heal. The next trainer stands just south of here.
Quick Claw. Go left all the way and into the classroom. The little Preschooler girl in front of the classroom on the left will battle you.
Oran Berries. Now you can go back outside and talk to the boy by the battlefield and he'll battle you.
TM01 (Work Up).
Great Balls. Kukui arrives after your battle and introduces you to Ilima, the first Trial Captain you will encounter in Alola. He runs his trial in Verdant Cavern. Ilima looks forward to your challenge. For now, follow Lillie into Hau'oli City.