If Cubs fans didn't care, they would shrug it off completely, but I have friends who love the Cubs that bring it up constantly.
Twins are definitely the primary rival - there was legitimate hatred between the teams, especially when Mauer and Morneau were around.
#3 kind of rotates between Indians, Tigers, and Royals in that order, whoever is best at the time. If no one is doing particularly well, that's the default order based on history.