Which was the first rock band?

Bill Haley and the Comets

1952
Bill Haley Saddlemen become the Comets
1953
Bill Haley's "Crazy Man Crazy" is the first rock and roll song to enter the Billboard charts.
1954
Bill Haley's "Rock Around the Clock" is the first rock song used in a movie soundtrack

I also found this:

According to music historian Peter Guralnick the first rock and roll artist was by Jackie Brenston and his Delta Cats who recorded Rocket 88 for the Sun Records label in 1951. https://en.wikipedia.org/wiki/Jackie_Brenston
Some will consider him a solo artist, some will consider it a group.

I believe "Rock Around the Clock" was the first rock song to go number one.

I think determining who the first rock band was is really a matter of perspective.

Since it was more of an evolution of southern blues, you can see in the timeline link that references to rock and roll are made as early as 1938.

The white guys became the mainstream for music that poor black folks had been singing for years. Not taking anything away from Bill Haley, Jerry Lee Lewis, or Elvis...but it was simply a style that had evolved and grew in popularity.