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I am looking for a clear, self-contained text (either a book or lecture notes) that deals with characteristic classes, starting from the very basics (fiber bundle, principal bundle etc.), and preferably dealing with the special examples of Steifel-Whitney class and Euler class.

Thanks!

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Milnor's book on characteristic classes is almost surely exactly what you want.

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    $\begingroup$ Don't forget Stasheff! $\endgroup$
    – user64687
    Jan 15, 2014 at 13:50
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Bott and Tu is quite self-contained.

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