I want to move the camera inside a 3D scene like this in a first person shooter style, how can I do that?
Your questions is rather general so I'm going to point you over to the great NeHe tutorials which can explain this far better than I can in the space allowed for SO Answers...
http://nehe.gamedev.net/data/articles/article.asp?article=08
This topic can get very in-depth but the above article is a good start at understanding matrix transforms and how it interacts with a "camera".
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