Why does sound move faster in solids? I know that the molecules are closer together in solids, and I know thicker springs also respond carry waves faster than thinner springs, but for some reasons I can't understand why. 
The molecules will have a larger distance to move before colliding with another molecule, but in a thicker medium wouldn't that time just be spent relaying the message between multiple atoms? Why is the relaying between a lot of tight knit atoms faster than one molecule moving a farther distance and colliding with another?
 A: I assume "faster in solids" means faster than in gases.
The speed of a mechanical wave is in general proportional to $\sqrt{k/m}$, where $k$ is some measure of the restoring force (e.g., the tension in a string, or a Young's modulus), and $m$ is some measure of inertia (e.g., the mass per unit length of a string, or the density of the medium).
Compared to a gas, a solid has a density that is greater by about a factor of $10^3$. However, gases are very compressible, while solids and liquids are extremely incompressible. The incompressibility factor overwhelms the inertial factor.
A: The difference between solids and gases appears in the momentum conservation equation:
$\rho\frac{d\vec v}{dt}=\vec S$ where $\vec S$ is a source term that expresses the rate at which momentum is exchanged between neighboring volumes, a "restoring force".
In gases, $\vec S=-\vec \nabla p$, where the pressure $p$ relates to density and temperature through equations of state, which describe the interaction between gas molecules. In statistical physics, this term is written as a collision operator. It means that the transmission of the disturbance, or the restoring force, is driven by collisions between molecules. These collisions are rare in average, so each molecule travels a long way before transferring its momentum.
In solids, $\vec S$ is the electrostatic force on an atom that is displaced from its equilibrium position. A very small displacement is enough to imply a large force because of the proximity of other atoms repelling each other. Also, the distance at which atoms "feel" each other is generally higher in solids than in gases (charge interaction instead of dipole interaction). Overall, atoms in solids don't need to travel long before they transmit their momentum.
That's why the disturbance propagation is faster in solids than gases.
A: Think of it this way. Elasticity is a property of material that allows it to store energy and release it without dissipating. Solids have high elasticity, therefore, they can store and release energy quite efficiently. Liquids and gases have low elasticity. They are also viscous and dissipate energy instead of transmitting it. Please note that I am not connecting speed of waves to viscosity but merely pointing out its dissipative nature.
