After helping Alder with some training and receiving your Medal Box, you find out about a new Gym Leader in Aspertia City. You should go there!
Visit the Aspertia Trainers' School
If you talk to people around Aspertia City, you find out that the new Gym Leader is Cheren. If you played Pokémon Black 2 and White 2, I'm sure you remember him as one of the rivals!
Inside the trainers' school, you find out that you will be battling some Normal type pokémon to get the badge. One kid in the school mentions that Riolu is a fighting type, which is strong against Normal type, and you can catch Riolu in Floccesy Ranch.
If you go to the back of the trainers' school, Clyde gives you a fresh water, then Cheren will notice you and say that you have to defeat the girl and the guy before Cheren will battle you.
The kid in the red cap is Youngster Pedro. He has a level 9 Patrat and a level 9 Lillipup.
Lass Serena is next. She has a level 9 Lillipup and a level 9 Patrat.
Gym Leader Battle: Cheren
Cheren has a level 11 Patrat and a level 13 Lillipup.
When you win, Cheren gives you the Basic Badge, as well as TM83 Work Up.
After you get your first Gym Badge, go outside. Bianca will be there. She gives you TM27 Return, a move that works better if you have a good relationship with your pokémon.
Cheren comes out of the gym and you all register each other in your Xtransceivers, and Bianca registers Professor Juniper for you as well. Your Xtransceiver will ring and you'll all talk for a little while.
Bianca says that if you call Professor Juniper, she will analyze your pokédex's completeness, and tell you about pokémon evolution.
If you call Bianca, she will tell you how well you and your pokémon are getting along.
If you call Cheren, he will tell you about pokémon abilities and type matchups.
Hugh walks up and challenges Cheren to a battle, and Cheren says to meet him in the school. Hugh gets all angry at Cheren's "weak" answer and rushes into the school.
Receive the C-Gear
Now Bianca will give you the C-Gear and will show you some of the features.
Bianca mentions Virbank City, which is just past Floccesy Town and has a Gym.
Before you go, first go into the house east of the stairs leading up to the Aspertia City Outlook (where you first found Bianca). Talk to the blonde girl upstairs. She will give you an Ultra Ball now that you have a gym badge.
Now you should go to Route 20, which is just past Floccesy Town.