My advice: don't get a degree in web design. DO go to college and learn about something you're interested in (or just go to figure out what you're interested in!). College should not be vocational training, no matter what your parents or counselors imply. Take the time to think really hard about things that have no bearing on the 'real world'. Keep designing websites (there will be plenty of opportunities) and build a portfolio. The best people practicing what you're interested are working and sharing what they learn online - not through academic institutions.

Some of the interactive design programs out there do look pretty good (SVA, for one - though that's postgrad).

(PS - I struggled with this 10 years ago and ended up getting degrees in computer science and graphic design. Anything I learned about 'web design' was totally useless by the time I was working, and I had taught myself basically everything I used day to day. However, the fundamentals of graphic design and CS do not change.)