Forget it. I'm just going to give up on my dream. I was right about the mistake I was making. I was trying to install Realtek sound card drivers, when I should have been scrolling down to download the ATI Graphics card drivers (R2.70).
Even so, the drivers installed but did not work. The sound control panel said the device was not plugged in and the Catalyst Control Center said my display did not support sound. I tried various things, but nothing worked.
In the end, I reverted back to the HDMI Audio drivers included in the Catalyst drivers and got sound, but unfortunately those crappy drivers don't provide an equilizer like the Realtek ones do.
So much for my upgrade. I got this card because my old Radeon was dying, and now I'm losing functionality that I used to have.