To some surprise when running a clean install of Ubuntu 13.10 (well, the Xfce Xubuntu 13.10 flavor to be exact) using the latest packages as of today, there still is no 2D/3D acceleration by default for the Radeon HD 7000 series graphics cards. The high-end AMD GPUs are simply falling back to the LLVMpipe Gallium3D driver for fallback CPU-based OpenGL acceleration of the desktop.
The Radeon HD 7000/8000 series support on the open-source driver is developed via the "RadeonSI" Gallium3D driver. While it took a while to be raised since the hardware enablement wasn't simply an extension of the Radeon "R600g" driver, it's stable now with Mesa 9.2. Additionally, the RadeonSI GPU support requires the AMD GPU LLVM back-end that was merged into LLVM 3.3. There's still a lot of RadeonSI work needed but if using the latest code it's in decent shape. The DRM/KMS support has also been present for some time now in the mainline kernel.
Ubuntu 13.10 has Mesa 9.2, LLVM 3.3, and the Linux 3.11 kernel, which together makes for an acceptable level of open-source Radeon HD 7000 series hardware support. The problem in the Ubuntu 13.10 stack is that it still doesn't package GLAMOR. The AMD HD 7000 series and newer hardware doesn't implement 2D acceleration via EXA in the xf86-video-ati DDX, but requires the GLAMOR library for running 2D commands via OpenGL.
While GLAMOR has been available for months in released versions, it's still not packaged and found in the Ubuntu 13.10 archive. This is rather silly especially with this being a blocker to having open-source 2D/3D hardware support on modern AMD Radeon GPUs. We're now past the Ubuntu 13.10 feature freeze but one of the bug reports relating to the lack of RadeonSI support can be found on Launchpad.
GLAMOR also isn't found in the xorg-edgers PPA but can be found in the Oibaf PPA. The issue with the Oibaf PPA is that Ubuntu 13.10 isn't yet supported with 13.04 being the latest stable version.
Long story short, if you want to try out the open-source RadeonSI support on Ubuntu 13.10 for now you will need to build and enable GLAMOR from source as detailed on the FreeDesktop.org Wiki. It doesn't look like Canonical will enable GLAMOR / RadeonSI support by default in Ubuntu 13.10. If you're looking for a Linux distribution with the best open-source Radeon Southern Islands support, I would recommend trying out Fedora 19 or the latest development packages for Fedora 20. Red Hat has been shipping RadeonSI support in Fedora for months.