Use the 'forward' panel on the giant stone D-pad to raise the dragon's head one level and reach a higher trainer.  The 'down' panel retracts the head downwards, eventually enough so that you can leave the Gym.  The 'left' and 'right' panels move the head in their respective direction.  Use these controls in conjunction to get where you want to go and face who you want to battle.

On the lower left foot is the defense-oriented Rhona.

Battle: Veteran Rhona (3600 Poke)
Opponent: Fraxure Lv 45, Druddigon Lv 45

Rhona is very heavily invested in her defensive methodology.  Her Fraxure loves to alternate Swagger and Protect until it's forced its first foe to knock themself out, then swapping over to Double Hit and Dragon Claw to finish the job.  On the other side is her Druddigon, which combines its Rough Skin ability with a Rocky Helmet to damage physical attackers twice over each time they come in contact with it.

On the lower right foot is the offensive Jerry.

Battle: Veteran Jerry (3600 Poke)
Opponent: Fraxure Lv 45, Druddigon Lv 45

Jerry's focus on offense is a bit less clear- while his Fraxure uses Dragon Dance to boost its attack and speed repeatedly, it also has the ability Mold Breaker, which allow it to hit targets regardless of typing- this means that even a Shedinja will be struck by its Normal attacks, provided it bothers using them instead of Dragon Claw or X-Scissor.  Druddigon is even worse, using a Dragon Gem to boost its first attack, invariably a Dragon Claw, to obscene power levels and then packing Flamethrower to send the inevitable Ice and Steel pokemon that will come out against it packing.