So have to ask what a solid is to answer this question.

Sticks are quite complex, so lets consider a simpler solid: an elementally pure iron rod.

You can imagine said rod as if it were a fixed array of crystalline atomic cores surrounded by a jelly-like substance. In this 'jellium' model the atomic cores have a positive charge, they are the protons and neutrons, and the jelly has a negative charge. The jelly is the wavefunction that represents the electron structure in bulk. If that makes no sense, congrats on knowing your limits.

You've probably seen the more modern model of an atom where there's a nucleus and around it is an electron fuzz with discrete energy levels. Or if you've studied at uni strange geometry representing a threshold in probability of finding the electron/s there on a given measurement (if not familiar under certain conditions reality kinda unfocuses it's eyes and things that we often think of as points become volumes of possible effect). This is a good model of a single atom, but when we bring atoms together they change each other's properties and the result is that these density functions (the weird electron cloud/shape things) start to blur together.

In our iron rod the electrons delocalize sufficiently we can kinda think of it as a weird jelly. A real stick is more complex, but can kinda be thought of as a stack of smaller jelly treats packed against each other.

When you push on the rod you're mashing the jelly of your hand into the jelly of the rod, this causes a shockwave that begins to spread, it propagates like a ripple in a skipping rope or a bounce on a trampoline. But since it's moving 'amount of electron like properties here'. That makes some areas more negatively charged which drags the positively charged atom cores slowly after it. It moves much slower than the speed of light as we aren't considering individual electrons which can move energy between them via photons, but the propagation of a disturbance in the collective arrangement of many that are tightly linked (we say coupled).

We can't imagine a stick that is perfectly rigid because we would be proposing a kind of matter that does not exist, one which isn't made of a lot of fuzzy electron jelly stuff but something else entirely. We can imagine matter where the jelly is very stiff, and consequently less energy goes into wobbling it all about and the squish moves forward very fast but that speed is still much slower than light because of this collective behaviour.