This is surely a pretty ignorant question, but I'm just starting out learning physics. In quantum theory the evolution of a particle's wavefunction is supposed to have two stages: deterministic evolution according to the Schrödinger equation when the particle is not being observed/measured, and nondeterministic collapse to an eigenfunction when it is being observed/measured. If this is the case, how have we experimentally verified that the Schrödinger equation holds for the deterministic stage?
Nice question. Let's split this into three parts:
(1) The Schrödinger equation has features that exist more broadly as fundamental principles of quantum mechanics. Briefly, these are: all information about the state is in the wavefunction; linearity; inner product; self-adjoint observables; unitary evolution; completeness.
(2) The Schrödinger equation has specific features that are not more generally valid in quantum mechanics. For example, it's nonrelativistic.
Taking these in order:
We see no evidence that any of these are violated, but we also don't have any good ways of singling out any particular principle from this list and testing it quantitatively. To do that, we would need a test theory that differs from quantum mechanics but is viable, and right now nobody has any idea how to construct a useful test theory. The trouble is that there are no-go theorems showing that quantum mechanics is very brittle. That is, if you try to introduce any violation of these basic principles, no matter how small, the whole thing seems to break down. Some material about this is given in Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," http://arxiv.org/abs/quant-ph/0401062 .
We find, for example, that the Schrödinger equation predicts the energy levels of the hydrogen atom pretty accurately. The errors are at about the level we expect based on the fact that the Schrödinger equation is nonrelativistic, and an electron in a hydrogen atom has $v\sim0.01c$. We can verify various other effects in the Schrödinger equation, such as tunneling and the behavior of the two-state system.
The part about collapse isn't actually part of the theory, it's just a feature of the Copenhagen interpretation. The Copenhagen interpretation is optional. There is absolutely no evidence for any actual physical process resembling the collapse described by the Copenhagen interpretation. All actual observations can be explained in other ways, e.g., in explanations based on decoherence.
Rephrasing your question as: "How do we experimentally confirm the deterministic evolution of the wavefunction (when measurements require collapse of this wavefunction)?"
This is actually very simple. We repeat the same experiment over and over and see if our output probability distribution matches what we expect our wave-function to deterministically evolve. If we get a probability distribution that doesn't match the (square of) wavefunction, then the theory is wrong.
You might be wondering "Does collapse get in the way of really truely knowing what the wavefunction was before it was measured"? It does, and that's why we redo the whole experiement after each time we measure the state.
If you're wondering if maybe somehow the collapse affected the earlier deterministic evolution, look at the example of a quantum Mach–Zehnder interferometer. You take a single photon, split it into two possible directions, and then recombine them again. You can see that when recombining the states, quantum interference controls which of the two output paths the state travels through. You can infact get the quantum interference to cancel so perfectly (by changing the difference in distance of the two paths) that you can completely control if the photon exits one of the two output paths or the other path. Here the deterministic evolution of the wavefunction is interfering, producing a completely deterministic, measureable outcome. So you don't have to have any concern over the indeterminism of quantum mechanics getting in the way of observing the determinism!