Hobbies And Interests
Cherrygrove
You fight your rival for the first time as you're returning from Cherrygrove City after meeting Professor Oak. He's stolen a Pokemon from Professor Elm and challenges you with it. The Pokemon he's chosen depends on the one you picked at the beginning of the game: he'll have Chikorita if you picked Totodile, Totodile if you picked Cyndaquil, and Cyndaquil if you picked Chikorita. Though he will have a type advantage on you, it is nothing to be worried about: his Pokemon has no special attacks. So let loose with your Normal-type attack and heal with Potions if your Pokemon's HP gets low.
Azalea Town
You'll next fight your Rival in the gatehouse west of Azalea Town, after rescuing all the Slowpokes. His first Pokemon has by now evolved and reached level 18, and he also has a Zubat and Gastly along too. Gastly is a Ghost-type Pokemon that is immune to Normal-type attacks, and its "Levitate" ability makes it immune to Ground-type moves like Magnitude as well. Psychic-type and Flying-type moves work well against it. Zubat can be beaten with Electric or Rock types. Ground and Rock types work well against Quilava, Electric and Grass moves are strong against Croconaw, and Bayleef will go down to Flying, Bug and Fire type moves.
Ecruteak City Burned Tower
Your next fight comes two towns later, in Ecruteak City's Burned Tower, as you climb the tower. He's caught a new Pokemon Magnemite, but his levels has only gone up a few points -- his starter is at level 22, for instance. Magnemite is an Electric/Steel type, resistant to Normal-type moves, super weak to Ground-type moves and also vulnerable to Fighting and Fire. Use the same strategy you used before to take out his other Pokemon.
Underground Basement
You won't fight your Rival again until Team Rocket takes over Goldenrod City. When you do, find him in the Underground Basement. His levels have by now increased quite a bit, with Zubat and Gastly evolved to Golbat and Haunter, and his starter is in its final form and level 34. Lastly, he has a new Pokemon, Sneasel, a Dark/Ice type. But you should be a lot stronger too. Sneasel is super-weak to Fighting moves, and the rest go down quickly to your strategies from earlier.
Victory Road
The climactic battle with your rival comes right at the end of Victory Road -- he finally has a full team, having added a Kadabra to round out his weakness to Psychics and Fighting-types. Use Ghost-type moves to take it out. His starter is at level 40, but it's nothing you can't handle if you're planning to beat the Elite Four. Beat him here to have him repent for his earlier selfish actions and promise to reform.
Mt. Moon
You fight your rival once more during the main story, as he requests a rematch at Mt. Moon. Haunter and Kadabra have by now evolved to Gengar and Alakazam, and all Pokemon are at level 45-50, but as long as your own Pokemon are similarly leveled you should have no trouble. Be careful with the Gengar though; it has Shadow Ball which can easily trash a low-level Psychic type.
Rematches
After you get all 16 badges, you can rematch with your rival. To do so, you must first have a rematch with Clair and Lance -- a double-battle. Return to the Dragon's Den on Tuesday, Thursday or Saturday and enter the house to start that fight. Afterwards, you can rematch your Rival on Monday or Wednesday at the Indigo Plateau; go up the central staircase to trigger the fight. His Pokemon are mostly the same, though his Golbat has by now evolved into Crobat, and are level 55-60.