Upon arriving at Hulbury, heal up your Pokémon at the Pokémon Center. You will run into Rose running incognito in a rather...silly outfit, I should put it, alongside his secretary Oleana being hoarded by a group of fans. You also briefly encounter Bede talking to Rose before walking off, and then Rose will notice your presence. He encourages you to go to the Stadium and defeat Nessa, and to join him at the Captain's Table restaurant afterwards.
Now, since you're here, there's a bit to do before doing heading to Hulbury Stadium. Chief among these is an in-game trade: if you have a Minccino, you can trade on for a Cottonee!
A colored background means that the Pokémon can be found in this location in the specified game. A white background with a colored letter means that the Pokémon cannot be found here.
Hulbury Stadium
Before arriving at Hulbury, you will be met with a Gym Trainer who is guarding the entrance. You will be told that Nessa is at the Hulbury Lighthouse, so make your way there: it's at the easternmost part of Hulbury. Meet up with Nessa, and you will be given her League Card before asking you to challenge her at the stadium.
You can get a Lure Ball from the Ball Guy here. The second Gym specializes in Water-type Pokémon. Now for this gym challenge, you will be met with a puzzle that involves setting up colored switches that activate and deactivate water pipes: pressing a colored switch will activate inactive water pipes of the corresponding color and deactivate active ones of said color. You must press these switches in a certain order while battling Trainers along the way. After you have completed this puzzle, you will battle with Nessa. She will Dynamax her main Pokémon, Drednaw, at the first opportunity. Upon defeating her, you will receive the Water Badge and the TM for Whirlpool.