Instructions
Defeat the Mahogany Town Gym Leader, Pryce. He has a Level 27 Seel, a Level 29 Dewgong and a Level 31 Piloswine. These are all Ice/Water-type Pokemon, which are weak against Electric- and Grass-type attacks.
Leave the gym and Professor Elm will inform you that Team Rocket has taken over the Goldenrod City Radio Tower. Head west to Goldenrod City and battle your way through the Team Rocket Grunts to the top of the Radio Tower.
Speak to the man who appears to be the Radio Director at the top of the tower and he'll reveal himself to be a Team Rocket Executive. Defeat him to get the Underground Key -- he has five Level 30 Koffings and a Level 32 Weezing, which are particularly weak against Psychic-type Pokemon. Go down to the Goldenrod Underground and head all the way to the right. Use the Underground Key on the locked door here.
Head into the Rocket Underground and immediately you'll be faced by a puzzle made of switches. Set the switches from left to right as three, two and one. This will open the next room, where you can find the real Radio Director. He'll give you a Card Key which will open more doors in the Radio Tower. Head back up the Radio Tower and use the Card Key to go through the locked doors on the third floor.
Go up to the fifth floor to face a pair of Team Rocket Executives, one male and one female. The female Executive has a Level 32 Arbok, a Level 32 Vileplume and a Level 32 Murkrow, which you can beat with Psychic-, Fire- and Electric-type Pokemon respectively. The male has a Level 33 Houndour, a Level 33 Koffing and a level 35 Houndoom -- Houndour and Houndoom are weak against Fighting-, Ground-, Rock- and Water-type moves.
Head to Blackthorn City in the northeast of Johto and take on the Gym Leader, Clair, there. Her Pokemon are all Dragon-type; she has three Level 37 Dragonairs and a Level 40 Kingdra. Dragon-types are particularly weak against Ice-types and other Dragons. Defeat Clair and she will let you into the Dragon's Den.