There's a few Trainers for you to fight down here, but none of them should be that difficult compared to what you've just fought.

You'll need to cross the whirlpool in order to get to the entrance of the building, which is where you have to take the test thing. Cross the whirlpool and then, before entering the building, keep going east and you'll find the Dragon Fang, which powers up Dragon-type attacks.

Inside the building, talk to the elder there and you'll need to answer some questions. If you answer incorrectly, the elder will cough and look a bit disappointed, and force you to retake the question. That's nice, but you don't want to do that, because then your Dratini will be less cool! Yeah, you heard me.

You see, if you answer all of the questions correctly without messing them up, the Dratini he gives you will have ExtremeSpeed, a pretty nice move that always goes first. Otherwise, it'll just have the weaker Quick Attack.

To get the ExtremeSpeed Dratini, just answer the following questions based on their order:

1. Top
2. Top
3. Bottom
4. Top
5. Middle

That should result in him giving you the ExtremeSpeed Dratini. Awesome! If you don't have room in your party, you'll have to come back later with room for it.

After passing the quiz, Clair will come down and will give you the Rising Badge, which makes it so no traded Pokemon will disobey you, regardless of its level, and she'll also give you TM59 (Dragon Pulse), a powerful 90-power Dragon-type attack. If any of your Pokemon can learn it, I recommend teaching it to that Pokemon (or the one with the highest Sp. Atk if more than one can learn it). You'll really appreciate it later on.

Clair will then tell you that you need to head east of New Bark Town, on to Route 27 in Kanto, if you want to get to the Pokemon League and take on the Elite Four.

Sounds like a plan! Make your way out of the Dragon's Den and, after you leave, you'll receive a call from Prof. Elm telling you he has something for you in New Bark Town. You know where to go now! New Bark Town!