Within any family (group), how do both types of numbers change from element to element?

I assume that you are referring to the Atomic Number and the Mass Number of elements. The former is the number of protons held in an atom of a given element, while the latter is the sum total of protons and neutrons in an atom of a given element.
As we move from top to bottom in a group in the periodic table, both these numbers increase. The reason is the presence of additional electron shells as we go down. For example, in Group 2 (Alkaline Earth metals), Beryllium has 2 electron shells and an electronic configuration of 1s2, 2s2. The next element down the group, magnesium, has 3 electron shells and an electronic configuration of 1s2, 2s2, 2p6, 3s2. The atomic numbers of Beryllium and Magnesium are 4 and 12, respectively.
An increase in the atomic number down the group also results in an increase in the mass number as one travels down a group in the periodic table. For example, the mass numbers of Beryllium and Magnesium are 9 and 24.3, respectively.
Hope this helps.