While there are many Wild Pokémon to battle among the Dungeons, there are at least 3 Rival teams for you to go against

The Shiftry Possé

Shiftry and his two Nuzleafs go around causing mayhem, however early on when they kidnap a Jumpluff, they get wiped on the floor by a Zapdos!

Red Partnership

In Red, The Red Partnership aren't quite your enemies and do offer some helpful advice as you go along. They get beaten by Groudon and then stay in town to give you advice. They feature Charizard, Alakazam & Tyranitar

Blue Partnership

In Red, the Blue Partnership are on the evil side and actually steal some of your Mail to get jobs. You end up battling them early on, however they will crop up and cause problems along the way. They feature Gengar, Medicham & Ekans

New Partnership

A Minor Partnership set up by Shiftry and features Blastoise, Golem & Octillery. They too were defeated by Groudon and stayed in town afterwards