If you are considering a career as a web engineer, I think the first thing you need to ask yourself, do you enjoy tying and untying knots over and over again. If you do, you probably have the right stuff to be a really good programmer.
I wish I knew this before I started my Master's degree in IT. Programming is frustrating if you don't have an engineering background... you keep running into new problems and people telling you what you could do better.