Your suspicions of a problem with the graphics card is correct.
After the error messages will the system contine to a command line prompt? You may have to continue thru the error message box(s).
If so, enter;
Code:
dpkg-reconfigure xserver-xorg
This should take you thru a series of Q & A steps, when asked for device/drivers selecting VESA should provide a fix.
When finished;
This should get the GUI (graphical interface) running for you so you can "test drive" Ubuntu.
Since the live CD won't normally save session settings these steps will probably be required each time you run it.
The problem comes from the fact that ATI (and Nvdia) don't offer Linux drivers so you have to find and install the appropriate driver for your OS. Not that great a problem on a full install but next to impossible for a Live CD.
Check
here for an excellent source of help/information.
Also, a dual boot installation is quite easy.
This guide is highly recommended.
Hope this helps you out a little.