How to experimentally measure number of protons or electrons in an element?
Moseley measured the atomic number in 1913 with relatively modest equipment (by modern standards) but I doubt that you could reproduce his experiments in a typical kitchen.  

Atomic number at Wikipedia

Expansion as suggested in comments.  

Here is a quote from that article of what he did.  

"Moseley measured the wavelengths of the innermost photon transitions (K and L lines) produced by the elements from aluminum (Z = 13) to gold (Z = 79) used as a series of movable anodic targets inside an x-ray tube."

It might be just possible at home if you really knew what you were doing but I doubt that it would be cheap or safe.  What is regarded as safe has probably changed a lot in the last 100 years.