In Mt. Coronet, you'll find Cyrus ranting about perfecting the world before leaving. Collect all the items you can, and simply walk to the other side of this small section of Mt. Coronet, then out of the exit to Route 208. You won't be able to do anything important here until much later in the game.
There are many Trainers to battle on Route 208, however two of them cannot be fought until much later in the game, when you have the necessary HMs. Go across the first section of the bridge, make a quick detour north to grab a Great Ball, then continue east down the stairs, and through the tall grass, and you will be at the entrance of Hearthome City.
Also on this Route is the Berry Master's house. Each day, he will give you a fresh Berry. You may also get the Berry Searcher Pokétch app that shows where there are Berries ready to pick. His wife sells Mulch, which affects the growth of Berries.
On arriving at Hearthome City, a lost Buneary will run into you, shortly followed by its Trainer, who will thank you for catching her Buneary, and suggest that you visit the Super Contest Hall (the large building near the Hearthome Gym). Once you get there, you'll find her talking to your mom. They will both talk to you about Super Contests, and leave. Go to the front desk and talk to the woman dressed in purple. She will reveal herself as Fantina, this city's Gym Leader, and return to the Gym where you will be able to challenge her.
After that, you can take a look around the city. Next to the Pokémon Center is Bebe's house, where Sinnoh's administrator of the PC box system lives. If you talk to her and have a spare slot in your party, she'll hand over an Eevee. If you're into Pokémon breeding, consider saving your game before you receive the Eevee, since there is an only a 12.5% chance it will be a female and otherwise, you'll have to wait until you have the National Dex to find a Ditto or another Eevee.
A colored background means that the Pokémon can be found in this location in the specified game. A white background with a colored letter means that the Pokémon cannot be found here.
East of the Poké Mart is the Pokémon Fan Club. Go inside and talk to the man with the hat inside. After listening to his rambles, he will give you the Poffin Case, which is essential for making Poffins. Poffins can be made in the building west of the Poké Mart, known as the Poffin House. Near the Poffin House is Mr. Goods. Although he claims he can't remember who he is or his real name, he has dedicated himself to seeking rarities. He gives goods to Trainers once they have met certain requirements. These items can be placed in the Trainer's Secret Base located in The Underground.
Next to the Pokémon Fan Club is the Pokémon Hotel. If you go to the second floor and talk to the woman with the Clefairy, she will give you a Shell Bell, which heals the user by 1/8 of its max HP when it inflicts damage.
At the northwest and northeast of the city are the entrances to Amity Square, where you can find some Accessories, an Amulet Coin, TM43 (Secret Power), and TM45 (Attract), but you'll need to make a trip from each entrance to get all the items. Keep in mind that you'll need a "cute" Pokémon to enter - your starter in any of its evolutionary forms will work. Additionally, each hut contains three teleporters that can warp you to a different hut. To access them, enter the hut but stop on the tile just inside the entrance. Take a step forward, right, or left to access the hidden teleporters. It is the only way to access the area where you can obtain the Spooky Plate. A chubby man in overalls goes around Amity Square every day. To reach him, players must go through the east gate and use the teleporters. When found, he will give the player five Berries or one Accessory.
Hearthome City Gym
Once you're done exploring the town, head to the Hearthome Gym, located in the northeast corner of the city. The Gym leader, Fantina, uses Ghost-type Pokémon, which are weak to Dark- and Ghost-type moves. Ghost-types are also immune to Normal- and Fighting-type moves, but Normal-type Pokémon are immune to Ghost-type moves in return. The gym's structure is two darkened rooms each with a maze of Duskull lamps. Although you're given a flashlight at the entrance, it's still easy to see everything on the screen. The goal of the puzzle is to find a blue tile, note down the shape on it, then go to the door that has the same pattern. If you pick the wrong door, you'll find yourself back at the beginning (note that if you leave the gym and reenter, the symbol on the blue tile is randomized, so it probably won't be the same). After the second maze, you'll find Fantina - challenge her to a battle to earn the Relic Badge.
Typically of Ghost-types, Fantina's Pokémon are all pain to deal with, particularly her Mismagius. If you chose Turtwig as your starter or have added a Shinx to your team, their jaws should be able to Bite through Fatina's team with little problem. Her Duskull has high defense stats and may take a few hits to take down. It has Will-O-Wisp to burn and halve the Attack stat of an affected Pokémon, making it lose 1/8 of its HP each turn. Apply Burn Heals right away. It also has Pursuit and Shadow Sneak, Pursuit doing more damage if you try to switch out and Shadow Sneak always going first, like Quick Attack. Future Sight will do damage to any Pokémon, regardless of type, after two turns. Do not forget Future Sight, as it could take you out just as you think you're winning and have low HP. Contrary to he Duskull, her Haunter has very high Special Attack and Speed, but has no Special based attacks, and has quite low defenses. It has Shadow Claw, a Ghost type attack with a high critical hit ratio, and Sucker Punch, which will strike before you for 80 power, but only works if you are attacking that turn. Like Duskull, Haunter's role is annoyance, and it will disrupt you with Confuse Ray before going in for Hypnosis to set you up for Fantina's Mismagius. Mismagius comes packing the powerful move Shadow Ball, a Ghost-type attack that synergises with Mismagius' high Sp. Atk. Not good! It also has Psybeam and Magical Leaf backing it up, with Confuse Ray to disrupt as well. Staravia would do well here as it is immune to Shadow Ball and its typing halves the damage from Magical Leaf.
Once Fantina is defeated, she gives you the Relic Badge and TM65 (Shadow Claw). You can also use Defog outside of battle, which will come in handy later on Route 210.
Having earned your Relic Badge, you've now done everything there is to do in Hearthome. It's time for to move east to Route 209, however before leaving, consider purchasing Great Balls from the Poké Mart - they have a 50% higher catch rate than regular Poké Balls. Make sure your team is fully healed, because as you enter the Gate that separates Hearthome City from Route 209, you'll encounter Barry, who will challenge you to another battle.
This time around he'll have two new Pokémon that represent the types of the Starter Pokémon that he didn't pick at the beginning of the game. Additionally both his starter Pokémon and his Starly have evolved into their second forms. He shouldn't be that tough if you have a well leveled team, since your selection of Pokémon is much greater than it was last time.
From the entrance, go east, then north to travel directly to Solaceon Town, collecting any items that are of interest to you. On this route, there is the Broken Tower, which becomes the Hallowed Tower when an Odd Keystone (available from a Black Belt on Route 208, hidden in Twinleaf Town, or mineable in the Underground) is placed in it. After speaking to 32 people in the Underground, a level 25 Spiritomb will appear to battle you. You will pass the Lost Tower near the end of the route, but you will need to get HM05 (Defog) first for dealing with the fog on the higher floors. Fortunately, HM05 is available not far from here, so pass it by for now. Follow the path north to reach Solaceon Town.