How do I know if a ring I bought is really 14K gold or something else? i.e. 10K gold, 14K gold over silver, silver

You have to have it analyzed. It can be accurately analyzed non-destructively using an x-ray fluorescence analysis, or destructively (requires a small sample) using atomic absorption spectrophotometry. Jewelers use the specific gravity (density) of the metal. It's only approximate (probably ±10%).