After you have explored Eterna City and the Grand Underground, you will need the Eterna Gym Badge to proceed.
Talk to Gardenia
In Eterna Gym, Gardenia is standing in the doorway. Talk to her and she will go through the door. Follow her.
After a cutscene, you need to look for the hidden trainers. The first trainer is to the northwest of where you entered. You will battle Lass Caroline, who has a level 15 Cherubi and a level 15 Roselia. You get 540 Pokédollars for winning.
The next trainer is to the southwest. You will battle Aroma Lady Jenna, who has a level 13 Budew, a level 14 Budew, and a level 15 Budew. You get 780 Pokédollars for winning.
The next trainer is in the southeast of the gym, near where you entered. You will battle Aroma Lady Angela, who has a level 17 Turtwig. You get 816 Pokédollars for winning.
The next trainer is in the northeast of the gym. You will battle Beauty Lindsay, who has a level 17 Roselia. You get 1224 Pokédollars for winning.
Gym Leader Gardenia
Now you can battle Gardenia, who is on the green platform in the middle of the north part of the Gym.
Gardenia has a level 19 Cherubi, a level 19 Turtwig, and a level 22 Roserade. You get 2904 Pokédollars for winning.
The Forest Badge gives you the ability to use Cut outside of battle. Traded pokémon up to level 30 will obey you. You will also get the Stickers that Gardenia uses, and some TM86s, which teach Grass Knot.
Now that you have unlocked Cut in your Pokétch, you can visit some optional areas.
If you go west from Eterna City and enter Eterna Forest by the north entrance, you can go west and cut the trees, then go north. Get the Ether to the right of the building, then you can enter the Old Chateau.
If you go west from Eterna City, and go southwest in Route 205, you will reach some trees that you can cut down with Cut. If you go south from there, then west, you will find four berry trees.
South of those berry trees, you can talk to the person to get three Leaf Stickers B.
Go south from the person to find TM82 Sleep Talk on the right and a Silver Powder on the left.
Go west and you will find a Honey tree.
South of that, the path leads to more trees that you can cut, and then you can go south from there to reach the house on Route 205 where you can rest.
Go to the Team Galactic Building
Now that you can use Cut outside of battle, you can cut the trees that are blocking the entrance of the north of the Eterna City Pokémon Center. Before you go in, be sure to east from the building and cut the tree there, to open a path to TM46 Thief, which is north of the statue. Then, when you are ready, enter the Team Galactic Eterna Building.