Unlike Pokemon Red and Blue, your Rival has a few changes in Pokemon Yellow. For one, his Starter Pokemon has changed. Instead of having the superior Starter he would have in Red and Blue, he now has an Eevee. His team will then still stick close to the script of the other First Generation Games, but his Eevee will evolve into one of its three forms (Jolteon, Vaporeon, Flareon) dependent upon two early battles within the game.
Final Evolution Clauses For Rival's Eevee:
Jolteon: If you defeat your Rival in battle at Professor Oak's Laboratory and the optional battle on Route 22, your Rival will evolve his Eevee into Jolteon.
Flareon: If you defeat your Rival in battle at Professor Oak's Laboratory but lose or do not battle your Rival on Route 22, your Rival will evolve his Eevee into Flareon.
Vaporeon: If you lose to your Rival in battle at Professor Oak's Laboratory and lose or don't battle him on Route 22, your Rival will evolve his Eevee into Vaporeon.
Besides his Starter, your Rival is the same old provocative, polarizing figure he was in Red and Blue. And you will battle him a number of times - eight to be exact - throughout the main storyline:
Professor Oak's Laboratory:
You will first battle your Rival right after you have both been given your Starter Pokemon. Your Pokemon will be at the same level, and this is one of the battles that will decide the Final Evolutional fate of his Eevee. However, a player can use this to their advantage depending on which Eeveelution they'd like their Rival to have.
Rival's Pokemon Team: Level 5 Eevee
The next time you can battle your Rival will be on Route 22, just west of Viridian City. He has caught a new Pokemon and has trained his Eevee a little. This battle, too, will help determine what his Eevee will evolve into, and, aforementioned, a player can use this to their advantage.
Rival's Pokemon Team: Level 8 Eevee & Level 9 Spearow
Route 24:
The next time you battle your Rival will be on Route 24 heading up Nuggest Bridge, just north of Cerulean City. He has added two more Pokemon to his arsenal, and has leveled-up his Pokemon quite a bit since your last meeting.
Rival's Pokemon Team: Level 15 Sandshrew, Level 15 Rattata, Level 17 Eevee & Level 18 Spearow
S.S. Anne:
The next time you battle your Rival will be on the S.S. Anne directly outside the Captain's Quarters. His Pokemon Team remains the same as on Route 24, just at higher levels.
Rival's Pokemon Team: Level 16 Rattata, Level 18 Sandshrew, Level 19 Spearow & Level 20 Eevee
Pokemon Tower:
The next time you battle your Rival will be in the Pokemon Tower. Rattata is longer on his Pokemon Team, and he has two new, upgraded Pokemon.
Rival's Pokemon Team (Soon-To-Be Jolteon): Level 20 Sandshrew, Level 22 Vulpix, Level 23 Shellder, Level 25 Fearow & Level 25 Eevee
Rival's Pokemon Team (Soon-To-Be Flareon): Level 20 Sandshrew, Level 22 Shellder, Level 23 Magnemite, Level 25 Fearow & Level 25 Eevee
Rival's Pokemon Team (Soon-To-Be Vaporeon): Level 20 Sandshrew, Level 22 Magnemite, Level 23 Vulpix, Level 25 Fearow & Level 25 Eevee
Silph Company Building:
The next time you battle your Rival will be in the Silph Company Building. He has replaced his Fearow with Kadabra and his other four Pokemon have evolved, including his Eevee. This is the point in the game were his final team begins to take shape.
Rival's Pokemon Team (Jolteon): Level 35 Kadabra, Level 35 Sandslash, Level 35 Ninetails, Level 37 Cloyster & Level 40 Jolteon
Rival's Pokemon Team (Flareon): Level 35 Kadabra, Level 35 Sandslash, Level 35 Cloyster, Level 37 Magneton & Level 40 Flareon
Rival's Pokemon Team (Vaporeon): Level 35 Kadabra, Level 35 Sandslash, Level 35 Magneton, Level 37 Ninetails & Level 40 Vaporeon
The next time you battle your Rival will be on Route 22 (once again). Since your last battle, your Rival has added another Pokemon. This will be the last change, apart from a few Evolutions, that will happen with his team.
Rival's Pokemon Team (Jolteon): Level 45 Exeggcute, Level 45 Ninetails, Level 47 Sandslash, Level 47 Cloyster, Level 50 Kadabra & Level 50 Jolteon
Rival's Pokemon Team (Flareon): Level 45 Exeggcute, Level 45 Cloyster, Level 47 Sandslash, Level 47 Magneton, Level 50 Kadabra & Level 50 Flareon
Rival's Pokemon Team (Vaporeon): Level 45 Exeggcute, Level 45 Magneton, Level 47 Sandslash, Level 47 Ninetails, Level 50 Kadabra & Level 50 Vaporeon
Pokemon League:
The next and final time you battle your Rival will be at the Pokemon League. Since your last battle, your Rival has recently defeated the Pokemon League and has become the Champion of the Kanto Region. Two more of his Pokemon have evolved, and all of his Pokemon have gotten much stronger from Victory Road. At this point, your Rival (apart from yourself) is the strongest trainer you'll find in the game.
Rival's Pokemon Team (Jolteon): Level 59 Alakazam, Level 61 Sandslash, Level 61 Exeggutor, Level 61 Cloyster, Level 63 Ninetails & Level 65 Jolteon
Rival's Pokemon Team (Flareon): Level 59 Alakazam, Level 61 Sandslash, Level 61 Exeggutor, Level 61 Magneton, Level 63 Cloyster & Level 65 Flareon
Rival's Pokemon Team (Vaporeon): Level 59 Alakazam, Level 61 Sandslash, Level 61 Exeggutor, Level 61 Ninetails, Level 63 Magneton & Level 65 Vaporeon