There is a personal property tax in SC, as in some other states, which means you have a tax each year on your car. However, if you buy a car (at least, 6 years ago) as a SC resident you only pay $300 sales tax! As I recall, registration and other administrative fees are not expensive in SC.
You do need proof you paid sales tax on your car anywhere you re-register. Vermont made me pay it again on the then-current blue book value when I registered it there because I didn't have proof I paid NV sales tax eight years before.
SC is, all things considered, about average when it comes to taxes. They all get you one way or another. Here's a site that breaks out taxes by state:
Taxes by State (Vermont is the worst.)