If you want to make it completely from scratch, you'll need an oven that goes to about 1,000°C and can maintain that temperature for some time. This is needed to "bake" the high temperature superconductor that makes levitation possible.
Alternatively, there are places you can buy an HTSC. They aren't particularly expensive.
The next serious barrier is that even the "high temperature" SCs operate at very low temperatures. The only practical way to get these temperatures is with liquid nitrogen.
The rest is straight forward. You use the liquid nitrogen to cool the HTSC plate to bellow the critical temperature, and it becomes a type II superconductor. The cool thing about that is taht it can lock magnetic fields in place, allowing for leviation. (I don't know why they are calling it quantum levitation. It's a silly name. Yes, superconductivity is a quantum effect, but it's a bit like saying "quantum laser." They're all quantum.)
Anyways, there are some techniques on getting this to work with a very thin plate of HTSC, which allows for better lockign of the magnetic field, and the main reason we got all these videos floating about. But for basic effect, you just need HTSC, liquid nitrogen, and some magnets.
If you do manage to get liquid nitrogen, keep in mind that it's a very hazardous substance to work with. I would strongly recommend getting someone who knows what they are doing to actually handle it.
