Lifelong Cardinal fan here.
I certainly wish that the Card's and Reds were each other's greatest rival. Unfortunately, we are stuck with the Cubs. For the time being, it's a lousy rivalry. Under Dusty Baker, the Cubs are a bunch of over-confident, under-achieving, arrogant, classless, sloppy-playing, whining punks. Sometimes it's like playing agaist the junior varsity team of a mental institution. I think that 3 out of 7 of the Cub's suspensions this year have occured as a result of conduct during Cardinal-Cub games. It's really rather pathetic. They expected that the division would be their's from day 1. They can't handle it now that it's not. Cub fans are also quite a treat. They go from "wait 'till next year" to "Cub's Rule" over nothing but media hype. It's sad.
On the other hand, I've always considered the Reds to be a classy organization. I've enjoyed many conversations with knowledgable Reds fans. I consider the Reds and the Cardinals to be the greatest teams with the richest history in the National League. I've darn near been mugged by Cub fans at Wrigley Field (not a good place to be wearing a Card's jersey!). On the other hand, I've never had a problem at Riverfront/Cinergy (never been to Great American). Instead, I have enjoyed a friendly banter with the Reds fans that were seated around me.
I, for one, would love to see the Reds become our biggest rival.