Your Rival Returns
Now that you've kicked Team Rocket out of Azalea Town and earned the Hive Badge from its gym, you're ready to move on. If you haven't already done so, drop off an Apricorn with Kurt so he can make you a fancy Poke Ball. Remember he can only do one per day. Once you gain the fly ability later in the game, backtracking to him each day becomes much easier.
Take the west exit out of town to be greeted by your red-headed rival. He's still up to no good and ready to battle. What Pokemon he has depends on which starter you chose, of course. Since we chose Chikorita, he had a level 18 Quilava, level 16 Zubat, and level 14 Gastly. The team is pretty tough, especially compared to the Pokemon gym leader you just battled.
After his defeat, you're free to move on to Ilex Forest. This foggy forest follows a fairly straight forward path. Not far into the forest, you'll find your way impeded by a man who's complaining about losing his boss's Farfetch'd. It's up to you to find these two duck-like Pokemon and bring them back. You'll spot the Farfetch'd openly walking around on the screen (much like the Pokemon following you), and you'll need to sneak up behind them and press the A button to make the grab. This puzzle just requires you to step on one of the brown leafy patches to make a crunching noise, which causes the Farfetch'd to turn in that direction, and then you can sneak around the back.
Your reward is HM01: Cut. The Hive Badge lets you use this move outside of battle, so equip it to a Pokemon and use it to cut down any small trees blocking your way, and then you'll be free to move into the second half of the forest. Here you'll find a handful of trainers to battle, but no one as tough as Bugsy or your Rival. Other than the trainers, you'll also spot a man headbutting a tree. Talk to him to find out what his deal is, and he'll offer to teach the TM headbutt to a single Pokemon, which lets you headbutt trees to retrieve hidden items or wild Pokemon. Learn the move if you wish, and then continue north to leave the forest behind.
Route 34 to Goldenrod City
Route 34 is a small route flanked by grassy areas and puddles, along with six Pokemon trainers standing between you and your next destination. At the upper end of the route you'll find the Pokemon Daycare. The old man and his wife who run the place will "babysit" two of your Pokemon. Any Pokemon you leave with them will gradually grow in level over time, but be warned that the higher they grow in level the more it will cost you when you come to pick them up. This is also the game's breeding mechanism, since if you leave a compatible male and female Pokemon at the Pokemon Daycare, you'll wind up with an egg.
Continue north for one final trainer battle–against a Pokemon police officer–and then you'll be in the massive Goldenrod City. The city sports nearly a dozen buildings you can check out. Head to the massive department store to buy items not found in ordinary Poke Marts, play the mini-game at the casino for a chance to trade coins for rare Pokemon like Dratini, grab a free bike at the bike shop, or pay a visit to the radio tower. This last option is one you'll have to do for sure, since the radio tower is where you'll find Goldenrod's Pokemon gym leader.
Pokemon Gym Leader Whitney
Inside the radio tower you'll need to talk to the lady behind the far counter and answer an easy series of trivia questions about things you've learned so far in the game. Doing so earns you the radio card, which plugs into your PokeGear and lets you pick up radio broadcasts anywhere in Johto. For now this is merely an optional gimmick, but it will factor into the gameplay much later on.
Receiving the radio card draws the attention of a pink-haired girl, who comes over and strikes up conversation. This is Whitney, the city's gym leader, and she'll invite you back to the gym to challenge her. Follow her and you'll be well on your way to the third badge.
Whitney and her four underlings use Pokemon of the normal variety, which are weak against fighting-type attacks. Normal-type Pokemon are also notorious for their high hit points, which you'll discover when you're up against Whitney's level 17 Clefairy and level 19 Miltank. Miltank in particular can be a pain to take down, since it has the ability to heal itself. Perservere and use your most powerful fighting-type attacks, though, and you'll win the aptly-named Plain Badge. The Plain Badge lets you use the HM strength outside of battle.