Viola is the leader of the Santalune Gym. She uses Bug-type pokémon and will award the Bug Badge.
Go north after defeating Lass Charlotte. You will meet gym leader Viola. She first sends out a level 10 Surskit. If you traded a Bunnelby to receive a Farfetch'd from the hiker in one of the houses of Santalune City, now is a good time to use the Farfetch'd. Surskit is a Water/Bug type pokémon, so you shouldn't send a Fire type pokémon against it, because Fire is weak to Water. Quacklin' knows Aerial Ace, which is supereffective against Surskit.
Next, Viola sends out a level 12 Vivillon. Farfetch'd's Aerial Ace move is also supereffective against Vivillon.
Viola tells you to go through the back of the tent to reach the exit of the Gym. You'll come out a secret exit behind one of the big photos in the gallery. As you leave the gym, the guy near the entrance tells you to go to the Pokémon Lab in Lumiose City and show Professor Sycamore your first gym badge.
To get there, go west of the Trainers' School, then go north toward the dirt path. Viola's older sister Alexa (a journalist) will introduce herself and give you an Exp. Share. To use it, select it in your bag and turn it on. While it is turned on, all of the pokémon in your team will receive Experience Points after battles, even if they did not participate. Alexa goes north to return to Lumiose City, where she works at the big publishing house.
To go to Lumiose City, go north where Alexa went. You'll enter Route 4, Parterre Way.