Is the velocity with which molecules of the medium oscillate same as velocity of the sound in that medium? I wanted to know that under sound field applied molecules of the medium oscillate with some velocity in the direction of sound propagation. Is this velocity same as velocity of the sound in that medium? I am interested in liquid medium. 
 A: It can't be, for the celerity of "vibrating" fluid parcels / groups of molecules (for volume or surface waves) also depends on the amplitude of the wave.  Moreover $c_\phi = \omega/k$ ( and $c_g = \partial\omega/\partial k$ if dispersive) while the fastest parcels on the fastest moment of the cycle move at $A\omega$, without a dependency on k. 
Now at the true molecular scale, even without sounds wave around, the thermal agitation of molecules (not parcels) makes a distribution a velocity around 0 (if no ground motion like wind... or sound waves :-) ) with magnitude peaking around the order of magnitude of sound waves, at least for perfect gas. But it's a wide distribution (responsible for many macroscopic effect, e.g. evaporation/condensation, etc).
A: 
I am interested in liquid medium.

The answer is probably not because the speed of sound in most liquids decreases with increasing temperature and yet the molecules in a liquid would have more kinetic energy if the temperature increased. Source: Kaye & Laby Properties of sound in liquids
As ever water is anomalous and shows an increase in speed reach a maximum at about 75 $^\circ$C.  Kaye & Laby suggest the following reason "mainly because of the temperature dependence of the adiabatic compressibility of the water molecule itself"
A: I can elaborate on this question with respects to crystalline solids. Usually atoms in solids posses many possible vibrational modes. Information of all of them are collected in the phonon spectrum (see Fig. below).

Those vibrations are not oscillations of individual atoms, but their collective correlated motion. Atoms can vibrate in traverse or longitudinal directions relative to the direction of the wave propagation. These vibrations correspond to optical and acoustic phonons respectively. However, the sound waves are longitudinal waves only. Therefore, from many possible collective oscillations of atoms, only some of them are responsible for the propagation of sound. 
A: No. For linear acoustics, the speed of sound, $c$, is given by $$c^{2} = \left(\frac{\partial \rho}{\partial p}\right)_{\!s} \, ,$$ and therefore is related to the thermodynamic equation of state. (The subscript $s$ indicates constant entropy.) This is the speed at which the wave propagates through the medium. A typical value for water, for example, is $c \approx 1500 \; \mathrm{ m/s}$.
The amplitude of the velocity oscillation of the molecules is known as the particle velocity, $v$. For a plane wave, the particle velocity is proportional to the acoustic pressure $p$ as $$v = \frac{p}{\rho c}.$$ The denominator, $\rho c$, is called the characteristic impedance. Again for water, $\rho \approx 1000 \; \mathrm{ kg/m^{3} }$, so $\rho c \approx 1.5 \times 10^{6} \; \mathrm{ kg / (m^{2} \, s )}$. A modest acoustic pressure of 1 Pa would then have an associated particle velocity of $v \approx 6.7 \times 10^{-7} \; \mathrm{ m/s }$— Quite different from the sound speed!
You can refer to Chapter 1 of Acoustics by Allan Pierce for derivations and details.
