You must go through Ice Path to get to Blackthorn City. Ice path has some areas of slippery ice, as well as some puzzles involving rolling boulders with Strength into holes in the ground that can create the correct path for sliding on the ice to go through the cave.
Once you pass through that puzzle, you reach Blackthorn City.
There are a few things you can do here. There is a woman who will give you a Dodrio for a female Dragonair. While this isn't a very good exchange, you can at least get another Trainer ID to help increase your chance of winning something in the Lucky Chance drawing at the Goldenrod Radio Tower.
There is also the Move Deleter, who can delete all types of moves from a pokémon, including HM moves. There is the Move Maniac, who can teach any move to a pokémon who is at a high enough level to learn it, provided that you give him a Heart Scale, which can be found by Rock Smashing rocks. The old lady in the same house as those two will teach Draco Meteor to a dragon pokémon that has reached maximum happiness. Finally the Move Tutor will teach some powerful moves to certain types of pokémon. There is a book on the table that tells you what he can teach.
When you are ready, go to the Dragon Gym. Dragons are weak against Dragon moves, so beware! When you defeat Clair, she will refuse to give you the badge! What?! You will have to go to the Dragon Den behind the gym and talk to the Dragon Master. He will ask you some questions. Try to give the kindest, most cooperative answers, and you will pass. Clair will give you the much-deserved badge. You can go back to the old man with an empty space in your party to get a Dratini! If you answered all questions correctly the first time, the Dratini will know the ExtremeSpeed move.
After this is done, you will learn that Professor Elm wants to see you. Go to New Bark Town.