I am not asking for too many experimental details. Just give me a rough idea of what things could one measure using which instruments so that the ultraviolet catastrophe (Rayleigh-Jeans law) could be refuted. Also, how does one experimentally verify the Planck's radiation spectrum? Assume me to be a senior undergraduate while explaining things.
No instruments required. Turn on a light bulb and note that everything around it does not catch fire, vaporize, or turn cancerous. According to the Raleigh-Jeans law, an object with greater than zero temperature should emit more power at higher frequency light. However much power is emitted as visible light, more power should be emitted as ultraviolet light, still more as X-rays, and on and on. This leads to an infinite amount of power being radiated from all objects that are hotter than absolute zero. Obviously, this does not happen.
As for measuring the real spectrum of black bodies, it's a matter of separating the different wavelengths of light and using a photometer or photodiode to measure the power of each. The separation can be done with a diffraction grating and the measurements done at different angles, or with series of narrowband filters.