My initial install just used the svideo feed that went from my sat decoder to my old TV. The picture quality was roughly the same between old TV and new TV. The biggest difference was the ability to play DVD at 16x9.
The next step was changing my PC's MCE output from svideo to DVI. What blew me away was the sharpness of the text and general clarity of the entire screen. Also recognize that the video source was changed from 640x480 to 1920x1080 so the text was both much smaller and much sharper.
Third step was component video from the HD decoder to the TV which was then followed by HDMI between the two. I cannot recall there being much difference between between component video and HDMI. If there was it was less than the marginal difference between svideo into the old TV vs into the new TV.
The answer to the question of basic cable into a new generation TV may depend upon the TV, but in my case there were no compatibility issues and the picture was at least as good as before.
I suspect, as mentioned before, that the source artifacts will be amplified rather than subdued with the new TV. One obvious adjustment that should be made is to turn down the sharpness control all the way. There is no need to sharpen the source video unless, perhaps, you are playing your parent's family recordings from a VHS/Beta tape.
When you do get the HD feed, however, your tendency will be to want to turn sharpness contol up because now all the SD source material will look blurry when contrasted with the HD source.