I can help you a bit regarding Rio de Janeiro!

Flamengo - Fla is very similar to Corinthians, but in Rio de Janeiro. Largest fanbase, hated by pretty much everyone besides their own fans. I'd still say Vasco is their biggest rival even if the Fla-Flu rivalry is big these days, because historically the Clássico dos Milhões (Vasco-Flamengo rivalry) has always been the biggest derby in Rio. Their fans are not only associated to the working class, but to the favelas - it's common for members of the trafficking rings in the slums to be Flamengo fans.

Vasco da Gama - Historically a great club, but has been really struggling recently, with three relegations to the Série B since 2000. They're associated with the Portuguese, which also makes them a working class team. They are mantaining a very good unbeaten record against the other Rio de Janeiro teams for a while.

Fluminense - The rich people club from Rio de Janeiro. Like Botafogo, they don't have much history, but I'd say they've been the best club from Rio recently together with Flamengo. Their rivalry with Fla is pretty big these days, but in my experience, if you ask Flamengo fans, they still consider Vasco their biggest rival. Recently Fluminense fans have been in crisis mode because of the sudden transfer of Fred (yes, the 2014 WC Fred) to Atletico Mineiro.

Botafogo - Historically the smallest club of Rio, known to have less fans than the others, but they have a collection of great players along their existence, the biggest example obviously being Garrincha. Have been struggling recently with relegations, like Vasco.

Regarding rivalries: the biggest one is a toss-up between Vasco-Flamengo and Fluminense-Flamengo. I feel like if you asked an older fan, they'd say Vasco, but younger fans would say Fluminense because of Vasco struggling so much with staying in Série A. The other rivalries such as Vasco-Botafogo and Botafogo-Fluminense are pretty friendly though. Vasco-Botafogo is even called the "Classico da Amizade" (the Friendship Derby).