When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they are not I will be very confused.
Then I hear talk about "topological invariants of smooth manifolds". I think this is one of my difficulties. Does this means that if you have two smooth manifolds and a homeomorphism between these then the invariant is preserved and has nothing to do with the chosen smooth structure? It seems like a strange thing to say.
I believe that Milnor has shown that the integer Pontrjagin classes are NOT topological invariants, and Novikov has proved it is true that the rational pontrjagin classes are. I haven't seen anything indicating either way for Chern classes.
What about characteristic numbers then? The signature of a manifold is a topological invariant so certainly certain combinations of characteristic numbers can be, but is it true in general? The literature is usually a bit advanced so I don't think I could go through it easily but of course these are important basic questions, if anyone could clarify that would be great.
I would like to know most about Chern/Pontrjagin classes/numbers but if anyone has something else to throw in I'll gladly take it :)