After you receive the Devon Scope from Steven in Route 120, you can scare away the invisible Kecleon that was blocking access to the Fortree Gym.
Flying-type pokémon are weak to Electric, Rock, and Ice attacks, so make sure to bring pokémon that know those moves. Flying-type attacks are super-effective against Grass, Bug, and Fighting-type pokémon, so avoid bringing those types with you to this Gym. Flying attacks are weak against Rock, Steel, and Electric pokémon, so it might be a good idea to bring those types with you to the gym. Flying-type pokémon are strong against Fighting, Bug, and Grass moves, and they are immune to Ground moves, so don't use those types of moves against the pokémon in this gym.
In addition to pure Flying-type pokémon, you will encounter Steel/Flying (Skarmory), Normal/Flying (Swellow and Doduo), Water/Flying (Pelipper), and Dragon/Flying (Altaria) pokémon in this gym.
The Altaria is very weak to Ice attacks, so you might want to bring the Castform that you received in the Weather Institute. First have Castform use Hail, then have it use Weather Ball to do Ice damage. But be careful, because the effects of Hail will last for five turns even if you switch Castform out, damaging any non-Ice pokémon each turn.
From the gym entrance, go east through the rotating door, then up the stairs and north. Go through the left side of the revolving door if you want to avoid a trainer battle. Otherwise, you can battle Bird Keeper Jared here. He has Swellow and Skarmory.
Go north from Jared (the revolving door to the west won't move when pushed from this direction) and you can battle Picnicker Kylee, who has a Swablu.
Go west and south of Kylee, then push the revolving door to the right. Then go past Kylee again and go south through the gap created when you pushed the revolving door.
Go west and up the stairs. Push the revolving door up. Then you will battle Camper Terell, who has Doduo and Pelipper.
After battling Terell, go northeast and follow the path to Bird Keeper Will, who has a Dodrio.
After battling the Bird Keeper, go east and push upward through the small revolving door. You can battle Bird Keeper Bran, who has Pelipper.
After that, push downward on the big revolving door. Go to the right, and stand on the right side of the small revolving door, and push it left. Then go up through that small revolving door. Go up and left, and then go up the stairs to face Winona.
Winona has Swellow, Pelipper, Skarmory, and Altaria.
Against Swellow use Rock, Electric, and Ice moves. It is also immune to Ghost moves in addition to Ground moves.
Pelipper is very weak against Electric attacks, so those are the best attacks to use against it. In addition, Pelipper is weak to Rock moves. It is immune to Ground moves.
Skarmory is weak to Fire and Electric attacks, and is immune to Poison and Ground attacks.
Altaria is very weak to Ice attacks, and is weak to Rock, Dragon, and Fairy attacks. It is immune to Ground attacks.
After you Win
After defeating Winona, you get the Feather Badge, which causes all traded pokémon up to level 70 to obey you. It also allows you to use Fly outside of battle. This allows you to instantly warp to any location that you have visited, including Routes.
Winona also gives you TM19 Roost, which causes Flying-type pokémon to land on the ground and restore its HP by up to half of its maximum HP; but it loses its Flying type during that turn.
Walk south to leave the gym. Now you can continue to the southern part of Route 120.