Radeon HD 6300M, is in the evergreen family. It is considered a separate family from the rXXX families.
Generally it is always preferred to use the
radeon over the deprecated
radeonhd. There are exceptions, but unless you have a specific reason,
radeon is the preferred driver.
FOSS evergreen support is usable, but it's still not as mature as the r6XX and r7XX series cards. AMD has been actively working on FOSS drivers, to better support the newer families. As is the usual case, updates take time trickling down to distro repositories.
It is your option to consider using the official binary FGLRX catalyst driver, which is available from the AMD driver site or Debian non-free repositories. always try the repository first, if it does not work, then consider downloading the official AMD FGLRX installer. In the usual cases, the proprietary driver offers significantly improved performance. It also often has support for additional features, which might not be currently available in the FOSS
Either way, I would suggest you verify your acceleration is working as expected.
First check for OpenGL Direct Rendering, with this command.
glxinfo | grep rendering
## Expected response 'direct rendering: Yes'
Next check for XV Video Overlay, with this command
## Expected response, 'Adaptor #0: ...'
If you are missing these tools, they are available in the