Your Rival, or Gary
Gary is one step ahead of you the whole time. He always has just one more badge, or just a few more Pokémon, but somehow, you always end up beating him in the Pokémon battles he likes to challenge you to. Gary has spiky hair and a weird sense of fashion. He also has a bad attitude, and is known for saying "Yo! Smell ya later!" Gary will show up a couple times to challenge you, always with less than 6 Pokémon.
He's always bragging about how he has 50 different kinds, so why doesn't he carry 6 with him? I dunno.

Team Rocket
Trying to conquer the world, Team Rocket will show up everywhere in the game. You'll first spot them in Mt. Moon, trying to dig up fossils to sell at high prices. They also run Rocket Game Corner, the secret HQ of Team Rocket.
Rocket:"Don't touch the poster in the back of the room! There's no secret switch behind it!"

There are 4 underground floors to the Team Rocket building, but you need to take the elevator to get to the bottom floor. Once you're there, you'll have to fight Giovanni, the Team Rocket leader, who uses rock-type Pokémon and a Kangaskhan, which is not rock-type.
If you beat him, Giovanni will give you a Silph Scope so you can see ghosts in Lavender Town.