The pokémon that you should bring to Striaton Gym are Purrloin, which you can catch on Route 2, and the elemental monkey that you get from someone in the Dreamyard. Fill the rest of your party with the highest-level pokémon that you have.
How to Get Through the Gym Puzzle
To get to the gym leader of Striaton Gym, there are three curtains that you have to open. Each curtain has an elemental symbol and color on it. There are three buttons in front of each curtain, and you need to step on the one that represents the element that is super-effective against the one on the curtain.
For the fire curtain, step on water. For the water curtain, step on grass. For the grass curtain, step on fire.
Strategy for the Trainers
There are two trainers in this gym. One has an L11 Lillipup. I recommend using Purrloin against it, because it will be strong against Lillipup's Bite move.
The other trainer has an L10 Patrat and an L10 Purrloin. For the Patrat, I also recommend using Purrloin for defense against its Bite move. Against the Purrloin, any of your pokémon will do.
Strategy for the Gym Leader
Finally, you will meet the gym leader. There is not just one gym leader: there are three! However, you will only have to fight one. You will be pitted against the leader whose element is super-effective against your starter. However, with the elemental monkey that you received from the person in the Dreamyard, you should be in good shape.
Cilan is the grass type trainer, Cress is the water type, and Chili is the fire type. Each one has the elemental monkey that is super-effective against your starter, but weak against your monkey.
The leader has an L12 Lillipup that knows the move Work Up. This move boosts the pokémon's Attack and Special Attack stats. For this reason, be sure to take down the Lillipup as quickly as you can. Otherwise, its Attack and Special Attack will become so powerful that even your higher-level pokémon might be knocked out in one blow.
The Lillipup knows Bite, so again, Purrloin is your best choice against it because of its defense against Dark.
The Leader also has an L14 elemental monkey (Pansage, Panpour, or Pansear) that is strong against your starter. Fighting against the leader's elemental monkey shouldn't be too difficult if you use the elemental move of the elemental monkey that you received in the Dreamyard. Just be sure to keep it healed up. Also, keep in mind that the gym leader has at least one potion. This monkey also knows Work Up, so keep on the offensive.
After Receiving the Trio Badge
Once you have defeated the gym leader, you will receive the Trio Badge. Also, you will get the TM for Work Up. Unlike in past games, TMs can be used as many times as you wish! Sweet!
Now that you have a gym badge under your belt (or in your badge wallet or whatever,) the person who was blocking the stairs up to Fennel's room will move out of the way, so you can go meet with Fennel. Her house is to the south of the building to the right of the Pokémon Center.