I have an existing double switch for ceiling fan and light. There are three wires coming out of ceiling (white black and bare). The instructions with the fan say to connect the white to white, bare or green to green. And black to black. The fan has a black/white wire and they say to connect it to the wire (ungrounded) for the wall switch.

What wire should I connect the black/white to? White, black or bare?

This worked on an existing fan and light I have removed.

Take apart the switches, sounds like someone used the bare ground for the neutral, and maybe the black and the white for light and fan.
The switches will help sort this out.
You can add a receiver in the canopy which will use the black and white that you have to allow separate light and fan control.