I have only been out of HS for 10 years now and I am still young in my professional career. I actually went to college with a different career in mind (psychology) and ended up getting my BA in Computer Science at a State University in NY (SUNY). I fiddled with HTML and some ASP before hand, but never thought of a career doing it. So you are already ahead of me, haha. Now, I am a Lead Java Developer at my current company for one of our biggest projects.
Many points that were made in here are very true. Having a degree from a reputable college will definitely help you out. Businesses out there, especially software companies, know which places are good. I strongly agree that getting a BA/BS in Computer Science would be a great push toward getting a career in development. And since you are a whiz at math, go ahead and do a minor or second major in that. Like mentioned before, the concepts you learn in college are far more important than the language itself. For example, if you learn how a for-loop is used, then you should be able to write that loop in another language easily by simply learning the syntax.
Let me ask you a question to you though. Why did you decide on C++ as the language you are going to program in? Is it because you hear it a lot or because you want to do something specific with it? Do you want to continue with your web development (since you know advanced HTML) or do you want to write desktop applications? These are some things you may want to consider.
Once you figure out what you want to do, then do some research. If you go with Web Development, do you want to use JSP and Servlets? Do you want to learn JSF or ASP.NET? If you want to do application development, what about C#? Or Java? Or VB.NET? Of course, you always have the option of learning more than one of these, haha.
When you figure out what you want to learn, grab an IDE and PRACTICE! Find some tutorials or get a book and go through it. You don't need to wait to be done with college (or even HS for that matter). If you learn some now, you can be ahead of the game.
Last thing... I would recommend reading
Head First: Software Development by O'Reilly. This is a great book to learn software development concepts. And I would also recommend reading
Head First: Design Patterns.
But the best way to start a career in programming is deciding to do so and have the drive. Which it seems you have already.
I wish you the best of luck!