As you enter the Gym, Clyde will stop you and give you a Fresh Water. Gee, thanks, buddy. He'll show you that you have to fight using these giant stone dragon things in order to get up to the Gym Leader, and also that Ice- and Dragon-type attacks are effective against Dragon-type Pokemon. Cool. Be sure to refrain from using your Fire-, Water-, Electric-, or Fir- I mean Grass-type attacks against them, though, because they're resistant. Otherwise, pretty much any type of attack that isn't one of those will work well if you don't have Ice- or Dragon-type attacks handy.
Climb onto the dragon and run all the way across it until you reach the tail of the black dragon. You'll fight a Veteran there, and he uses a Fraxure L46. After beating him, walk backwards and then one of the arrows on the white dragon will activate. Step on that arrow and the dragon will rise upwards.
Three arrows now light up; the bottom one takes you back down, while left or right take you to some Trainers you'll have to fight. The one on the left has a Veteran with a Fraxure L45 and a Druddigon L45 on her team. The Druddigon has both Rough Skin and the Rocky Helmet equipped to inflict some massive recoil damage, plus it likes to confuse your Pokemon with Swagger and then use Protect. Very annoying! The Trainer to the right is a Veteran with a Fraxure L45 and a Druddigon L45 as well, although this one has the Sheer Force ability instead, so it won't annoy you quite as much.
You only need to beat one of those two Veterans in order for the Up arrow to light up when you go back to the center, but I recommend beating them both anyway.
Go up to the next level, then you have to fight at least one of two Trainers to move forward. The Veteran on the left uses a Druddigon L44, a Fraxure L44, and an Axew L44 in a Rotation Battle, while the Veteran on the right uses a Druddigon L44 (middle), Axew L44 (left), and a Fraxure L44 (right) in a Triple Battle.
All you need to do now is just go back to the center and step on the Up arrow to climb up and lock heads with Drayden's dragon. Heal up your Pokemon (taking a trip to the Pokemon Center if necessary), then get ready to take on the seventh Gym Leader in the game, Drayden!
He starts off with his Druddigon to get the fight rolling. This thing has several nasty attacks to hit you with, including Crunch and Revenge for some type coverage. Crunch hits extra hard due to Druddigon's Sheer Force ability, while Revenge makes any Ice-type pay if it doesn't knock out Druddigon in one hit. Its Dragon Tail forces it to strike last (actually, Revenge does, too), but it does decent damage and forces you to switch out afterwards.
His Flygon is the next thing up, and this thing is interesting in the fact that it's a Ground/Dragon-type with Levitate, meaning it is immune to Ground-type attacks (as well as Electric-type attacks, but you weren't going to use those against Drayden's dragons anyway, were you?). It packs Earth Power for doing potentially serious damage, as well as Crunch and Dragon Tail like its predecessor. Rock Slide can also be used to help combat Ice-type Pokemon looking to take advantage of its weakness.
Last comes his Haxorus. This thing is nothing to take lightly, because Haxorus has amazingly high Attack, plus it likes to open with Dragon Dance to boost both its Speed and Attack by one stage. That can quickly turn the fight against you, so try to take it out as quickly as possible! The only Dragon-type attack it uses, though, is Dragon Tail, which strikes last and gives you a chance to hit it regardless of its Speed bonuses, but it may decide to use either Slash or Assurance instead.
After beating Drayden, he'll give you the Legend Badge, which ensures traded Pokemon up to level 80 obey you without any hassle. He'll also give you TM82 (Dragon Tail), which, as you might have noticed, strikes last and pushes the foe away, ending a fight against a wild Pokemon or forcing an opposing Trainer to switch out after the damage is done.
There's no shortcut to leave, unfortunately, so you'll have to just keep taking the arrows down.