Speed of Medium Compared to Wave Speed

Do the particles in a medium oscillate about their equilibrium position at the same speed as the wave moves through the medium? Is it the same for both transverse and longitudinal waves?

No. The speed of oscillation increases with amplitude and frequency. This is true for both transverse and longitudinal waves.

There is another velocity: the speeds in random directions of molecules in a gas. These are (on average) larger than the speed of sound in that gas, but of the same order of magnitude.

Suppose a wave $y(x,t)=A\sin\left (2\pi f t - \frac{2\pi x}{\lambda}\right )$ where $y$ is the displacement of a particle from its equilibrium position which is a distance $x$ from an origin at a time $t$.
$f$ is the frequency of the wave and $\lambda$ is the wavelength.

The particle moves with simple harmonic motion, amplitude $A$ and frequency $f$ so the maximum speed of the particle is $2\pi f A$ and its minimum speed is zero.
The wave moves at a constant speed of $f\lambda$.

This is true for both longitudinal and transverse waves.