Start by teaching the columns first--this is the foundation of chemical interactivity. Each column on the periodic table showcases how full (or empty) the outermost shell of electrons is on that element. In general, if the outer shell is completely full (such as on the column starting with helium), the element is non-reactive. These are the "noble gases." If the outermost shell has only one electron in it (such as the column starting with hydrogen and working its way down through lithium and sodium--the alkali metals), the element is very reactive.
The groupings of elements (alkali metals, lanthanides, actinides, halogens, rare earths and such) are a great way to highlight other chemical properties. For example, copper, silver and gold are all excellent conductors. They're all in the same column on the periodic table, and they all have a number of other properties in common (such as malleability and oxidization reactions). The next column over contains nickel, palladium and platinum, which are all commonly used as catalysts, e.g. nickel plating, with palladium and platinum used in catalytic converters for cars.
Once you've explained how the columns of the periodic table highlight similar elements, it's time to discuss atomic masses and atomic numbers. Atomic numbers are based on the number of protons in the nucleus of the atom, and the elements are written in ascending order from left to right by atomic number, from hydrogen at 1 to seaborgium at 110.
Because the nucleus of the atom is made up of both protons and neutrons, atomic masses don't go up exactly linearly with atomic numbers. (The mass of electrons is so much lower than the mass of protons that they're negligible contributions to the mass of the atom). Highlight the difference in masses with the differences in atomic numbers. For example, tellurium is heavier than iodine, but lower in atomic number because it has an extra neutron.