Does a college course in computer programming allow one to work with web development?

My ideal career would be working with the internet -> Site development, maintaining websites, taking care of the e-commerce area, etc.

I am starting college in computer programming- Am i going in the right direction or does computer programming mostly involve developing software in general, with little of the internet involved?

If there are better college courses out there to work with site development, what would that be?

Thanks