Don't just copy headers, you will need to download the already built for Visual Studio Qt version (the one named Qt libraries 4.7.0 for Windows (VS 2008, 226 MB)), from here and the Visual Studio add-in(add-in doesn't work with express version of Visual Studio)
Or build Qt from source yourself.

But you can download the Qt SDK and use it, you have advantages with Visual Studio only if you already know the IDE. I think that you can separate the Visual Studio learning from Qt learning, and in the future "combine" them only if you really need them together.