is knot type invariant under diffeomorphism? Is it possible to have a diffeomorphism of $R^3$ which changes the knot type, for instance the image of a trivial knot is a trefoil knot?

Just to state that the answer given here by Emanuele depends on the fact that the isotopy classes of self-homeomorphisms of $\mathbb R^n$ , which is {$Id,-Id$}. If your not is embedded in a different space X, the answer will depend on the isotopy classes of homeomorphisms of X.  
Basically, we want to see if , for  h a homeomrphism , whether, $K=Id(K)$ and $h(K)$ are isotopic. But since MCG$(\mathbb R^n)= {\pm Id} $ , we have that h(K) is isotopic to $\pm Id K= \pm K$ .
