In QM, one takes the inner product $(\phi|\hat x|\phi)$ and $(\phi|\hat p|\phi)$ to compute the position and momentum expected values, but what does one do in QFT? What is the relationship between the wavefunction in QM and the QFT field?
For 1-particle states you can do exactly the same in (nonrelativistic) QFT, with position operator $X=\int dx a^*(x)xa(x)$. For multiparticle states, the notion of position makes no sense.
In quantum field theory, particle collision experiments are interpreted in terms of the S-matrix, which describes the transition from the far past, where particles are still described by separate, single-particle states, to the far future, where particles are again described by separate, single-particle states. In the intermediate time, close to the collision, one cannot identify individual in/out particles, and the system dynamics is that of a quantum field, not of particles.
In quantum field theory, the S-matrix is approximately computable in renormalized perturbation theory.