Great thread of discussion, throughout.
Welcome to the sport, OP.
As far as rider rivalries and being a fan goes, I am a fan of rivalries. So, for the classics, I am a fan of Boonen as well as Cancellara. Either's accomplishments would not be as significant if they did not occur (often) against each other. Cycling can be similar to boxing in this way, where a purist can be a fan of a great fight more than a fan of an individual fighter.
I'm also a fan of up-and-comers, or new splashes of talent such as the emerging Sagan and Kittel.
Being American, I usually like to see US riders do well, but was never a fan of Landis. My national loyalty for Lance waned after about his 3rd victory when it became apparent that everything wasn't on the up-and-up.
An interesting thing about cycling is that there are often intra-squad rivalries (Wiggins and Froome, Lemond and Hinault) that seem FAR more bitter than rivalries between teams. That said, there are national rivalries between teams from the same country. Here, Trek, Garmin, and BMC are rivals. This doesn't seem to play a role in many races, but it was notable when Garmin chose to reel in break containing Hincapie a few tours ago, thus depriving Hincapie from a yellow jersey.
One of the best things about being a cycling fan, IMO, is the rich history of the sport. For any new fan, I would suggest going to a local bike shop and asking for suggestions about some good reading about the sport's past.