I'm late to this, but perhaps I can shed some light...
Don't use someone else's data. That defeats the purpose. But aside from Io and Mercury, you can do a huge amount with a modest amount of equipment at home.
For example, for the blackbody spectrum, you need an empty kiln, a diffraction gradient (for separating wavelengths of light) and a photometer (for measuring the intensity of light). Heat the kiln to a temperature, put the diffraction gradient in front of an opening in it, and then measure the light intensity with the photometer at various angles from the grating to plot out the spectrum at that temperature. Change the temperature and repeat.
The original quantum mechanics work was based on emission lines of atoms. You need a tube with appropriate monatomic gases and a way of exciting it. They sell these for lab classes. Then you put current through the tube, put your diffraction gradient in front of it again, and measure the angle of the distinct lines you see. Those are exactly the data that Heisenberg was trying to explain.
A lot of other quantum mechanical predictions are made via statistical mechanics and describe the thermodynamic properties of material. Now you're back in measuring latent heat, specific heat, expansion coefficients, and all those unsexy things that you can do pretty straightforwardly.
There are lots of ways to measure the speed of light. John Baez describes some at http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLigh....
A lot of the stuff you hear about isn't feasible to reproduce because it's enormous or time consuming or expensive. You're not going to build another LHC (but you could probably build a little cyclotron and measure meson decays). You can certainly learn how to make observations of Mercury and measure its shift of perihelion over the course of the next couple years, but that's pretty low payoff for your time compared to table top experiments.
> Are there resources of others showing from experiment to theory with calculations of real data defending the theory's math
Yes. Experimental physics papers. Theoretical physics gets all the attention, but it's the minority of the actual activity of the field.