Though the Pickingverse is somewhat different, it's based off the games, so here is the rival's journey.

Pokémon Fire Red/Leaf Green

Starters + First Battle (backup)
The Pokédex (backup)
Gate of Victory Road
Nugget Bridge
The S.S. Anne
Lavender Tower (backup)
Silph Co.
Pre-Elite Four Battle
Championship Battle
Before Sevii Islands
Mid-Sevii Islands
Returning to Pallet Town

Pokémon HeartGold/SoulSilver

Cinnabar Island - Insufficient Badges
Cinnabar Island - Final Badges
Earth Badge
Fighting Dojo