I can't recall a particular Rose Bowl game from my early childhood. I typically watched the Cotton Bowl early in the day, and the Orange Bowl at night, which always had the "flashy" halftime show. For the Rose, I just recall that it always seemed to be USC playing Ohio State or Michigan. That might or might not be true for the years 1976-1983 when my football awakening was occurring, but that was my impression.
The Fiesta Bowl was not a major bowl until much later, I'd say the first one I remember was the 1987 Penn State-Miami game.