The easiest measurement is to take a ring or bracelet that fits comfortably and measure the diameter of the inside circle. Depending on the width of the band on a ring, you might need to order a size bigger by 0.5 to 1. With wider bands, you will want to go bigger.
For necklaces, take one that fits comfortably and measure with a ruler or measuring tape from one end to the other, including the clasp. This also works for bangles. Alternatively, you can use a tape measure to measure your wrist or neck.
For two-finger rings, you will need to know the sizes of both fingers.
Another alternative (if your piece does not have a gemstone) is to use baking soda with a little water, make a paste, rub it on the piece, then rinse thoroughly and dry. Your piece will sparkle.