Viewers didn't need new equipment. The analog broadcasting standard of the day allowed you to watch programs broadcast in color on a black-and-white set. In fact, the NTSC system approved in 1953 for color broadcasting in the US had been chosen with that compatibility in mind. Not until 1972 did color televisions outsell black and white sets.

The investment in equipment was all on the side of the networks and local affiliates, and only in 1966 did all three major networks have their entire prime-time lineup in color. NBC led, because of its connection to RCA, maker of much of the needed equipment. The network's famous peacock intro and "in living color" slogan touted its lead. Shows like "Bonanza" and "The Virginian" were known for and promoted their color broadcasts of scenic settings. Prime-time shows typically changed to color from one season to the next, as can be seen on reruns of "The Andy Griffith Show," "The Beverly Hillbillies," or "Gilligan's Island." A curious exception is the single color episode of "Perry Mason," shot and broadcast as an experiment just before the show was canceled.

The majority of prime-time shows were shot and edited on film, where color was already available. The bulky and power-hungry electronic cameras feeding videotape were used mostly for live events and variety shows, as well as soap operas and, starting circa 1970, some prime-time shows (like "All in the Family").

Some interesting tidbits about the transition here. Even more detail, from passionate amateur Ed Reitan, at this website.

When stations could colorcast, they didn't prize the b&w syndicated/off-net properties they had before, so shows like Jeannie and Bewitched converted to color filming in order to still be lucrative in the syndie marketplace. In the meantime, it was syndie shows recorded on film that suddenly did very well because their producers like Ziv had paid for color, so their backlog became suddenly very profitable.