ATI has now released a new free driver which works brilliantly on Debian. In the past, Debian users had to choose between using the community-provided free software driver, or the proprietary one. Generally the proprietary driver was more feature-rich, but the free driver worked more reliably across upgrades. So now, you can safely ignore old guides and start using it.
Make sure you are on Debian 9 (Stretch) or newer.
These steps apply to a fresh install.
You should use lspci to confirm that you have an ATI card.
$ lspci | grep Radeon 01:00.0 VGA compatible controller: Advanced Micro Devices, Inc. [AMD/ATI] Tahiti XT [Radeon HD 7970/8970 OEM / R9 280X]
You need to install a package called firmware-linux-free to get the driver working at all. If you want decent graphics performance, you will need firmware-linux-nonfree as well, which involves adding non-free sources:
Add the words “contrib non-free” to the end of your mirror:
deb http://.../debian/debian/ stretch main contrib non-free
Add the packages:
apt-get update apt-get install firmware-linux-free firmware-linux-nonfree
What, that’s it?
Well, yes, for a fresh install that’s it. If your install is old, you might also have to remove old drivers or install the xserver-xorg-video-amdgpu and xserver-xorg-video-ati packages (in my case, these were already installed).
The Debian Wiki AtiHowto contains some more detailed information, most of which is not relevant for a simple desktop setup.