Quantum computers are, unfortunately, quite hard to build. Experiments with polarizing filters or beam splitters would be able to demonstrate quantum effects, but I know of no way to make simple quantum circuits for multiple qubits unless you have single photon sources and detectors.
Alternatively, you could use current cloud-based devices. The IBM Q Experience has a simple GUI interface that would be suitable for students (after some introduction), and will then run the circuit on real hardware. If you students would be able to make circuits programmatically, they can use more quantum IBM hardware as well as hardware by Rigetti, with other companies also in the pipeline.
For a 'single qubit' experiment, you could perhaps just use polarizing filters. The $|0\rangle$ and $|1\rangle$ states of the qubit could be associated with horizontal and vertical polarization, and the $|+\rangle$ and $|-\rangle$ states could be associated with angles of $45^{\circ}$ and $135^{\circ}$. Then just by holding up a filter, you can turn sunlight into a stream of single qubits in a given state.
With a second filter, you can similarly measure in the $|0\rangle / |1\rangle$ basis (by holding it horizontally or vertically, and seeing if any light comes out) or the $|+\rangle / |-\rangle$ basis (by holding it diagonally). With multiple filters you could chain these measurements, and show the how the measurement bases are complementary. You could even remake the game I made to run on quantum computers: Battleships with complementary measurements.
This would be a single qubit example, despite the fact you have many qubits, because they are always the same state and they never interact. So you just have many samples of a single qubit process, that just happen to be shining down on you all at once.
Disclosure: I work for IBM, and Rigetti once gave me a t-shirt