Within any period, how do both types of numbers change from element to element?

I assume you are referring to the atomic number and the mass number of an element.
The atomic number is the number of protons in an element, while the mass number represents the sum total of protons and neutrons of an element.
In the periodic table, a period refers to a (horizontal) row of elements.
As one moves from left to the right across a period, the atomic number increases, since the number of protons is increasing. Each element has one more proton compared to the element to its left. For example, the order of elements (from left to right) in period 2 is lithium, beryllium, boron, carbon, nitrogen, oxygen, fluorine, and neon. The atomic numbers of these elements are 3, 4, 5, 6, 7, 8, 9, and 10, respectively.
Similarly, as one moves from left to the right across a period, the mass number increases as well. In period 2, the mass number of Lithium is 6.94, and it increases to 20.18 for Neon.

In summary, both the atomic number and the mass number increase as one moves across a period in the periodic table.