Let $f\in S_k(\Gamma_0(N))$, $(p,N)=1$ and $T_r:S_k(\Gamma_0(Np))\longrightarrow S_k(\Gamma_0(N))$ the trace map. Does then $T_r(f\mid V_p) = f\mid T_p$ hold?

You don't define your notations, as Olivier points out, but I'm going to assume that $V_p$ is the map $S_k(\Gamma_0(N)) \to S_k(\Gamma_0(Np))$ given by $f(z) \mapsto f(pz)$. Then the answer is "yes" (at least up to a constant factor, something like $p^{k1}$, that depends on your choice of conventions); in fact, this is essentially the definition of the $T_p$ operator  pullback along one degeneracy map followed by pushforward along the other. 

