The Dragon Club is stationed in Goldenrod City, accessible through its entrance in Goldenrod Tunnel. Membership to the Dragon Club is bestowed to anyone who has completed the Dragon's Den Quest, which allows access to the commodities and services of various NPCs in the club.
Points of Interest
TM02 - Dragon Claw
Herein, you can speak to the Club Manager (Justin) to purchase TM02 (Dragon Claw) for 2,000, after you have completed the Dragon's Den Quest and have been rewarded with club membership.
Dragon Pulse Tutor
Scientist Rosalinda, an avid researcher, is an expansive tutor for the move Outrage—if one completes the requisite mini-quest (explained below), she will offer to teach the move Outrage to a broad array of Pokémon.
For her research, she would like to observe the evolutionary mechanisms of Horsea, and — at a later stage in the task — Seadra. To aid her, she requests that you capture a Horsea, no higher-leveled than 35, then train it to level 99; she will require that it evolves through both of its stages, into Kingdra, for it to fully aid her research.
This mini-quest will also require club membership, thus completion of the Dragon's Den Quest is also required for this service.