Move Reminder: Mount Lanakila inside the Pokemon Center.
TM Shops: Heahea City Pokemart, Royal Avenue Pokemart, Koni Koni City Pokemart, Malie City Pokemart, and Seafolk Village Pokemart.
Move Tutors: Hau'oli City Shopping Mall, Big Wave Beach, Heahea Beach, Ula'Ula Beach, Poni Beach, and the Battle Tree.
Pokemon Nursery: Paniola Ranch, North of Paniola town.
Bottle Cap dude: If you mean the guy who provides Hyper Training, he's at the Hau'oli City Shopping Mall. If you mean the guy who will exchange Bottle Caps for shards, he's inside the castle of your Festival Plaza (post-game).
Source: Official guide-book, and I've played both games