There are many different Shops within Festival Plaza, each having a specific purpose. There are a total of 35 different Shops, many of which actually share the same functionality except with different results. To get more Shops to open, simply interact with avatars visiting your Plaza as they appear. After reaching Level 10, you'll be able to get Stores from these other players - this will allow you to obtain stores that are exclusive to the other game. Shops can be visited once per day.
List of Shops:
Bouncy House (Put your Pokémon on Courses to boost Stats): Clink-Clunk Land, Stomp-Stomp House, Phew-Phew Gym, Thump-Bump Park
Dye Shops (Dyes white clothing items to be a different color): Team Red, Team Yellow, Team Green, Team Blue, Team Orange, Team Navy Blue, Team Purple, Team Pink
Fortune Teller (Gives you hints of facilities you may be lucky in): Kanto Tent, Johto Tent, Hoenn Tent, Sinnoh Tent, Unova Tent, Kalos Tent, Pokémon House
Haunted House (Obtain an item): Ghosts' Den, Trick Room, Confuse Ray
Lottery (Play the lottery once a day to win items): Big Dreams, Gold Rush, Treasure Hunt
Restaurant (Get meals that boost your Pokémon's Stats): Rare Kitchen, Battle Table, Friendship Café, Friendship Parlor
Store (Purchase items): Ball Shop, General Store, Battle Store, Soft Drink Parlor, Pharmacy, Effort Shop