This is the way the periodic table should be presented.
The most important things to note are the use of the full name (so no one has to memorize the abbreviations to use the table), and the elimination of uncommon or radioactive elements (so no one is confused by things they'll never use).
I | II | IV | V | VI | VII | VIII | IX | X | XI | XII | XIII | XIV | XV | XVI | XVII | XVIII |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Hydrogen |   | Helium | ||||||||||||||
Lithium | Berylium |   | Boron | Carbon | Nitrogen | Oxygen | Fluorine | Neon | ||||||||
Sodium | Magnesium |   | Aluminum | Silicon | Phosphorus | Sulfur | Chlorine | Argon | ||||||||
Potassium | Calcium | Titanium | Vanadium | Chromium | Manganese | Iron | Cobalt | Nickel | Copper | Zinc | Gallium | Germanium | Arsenic | Selenium | Bromine | Krypton |
  | Strontium | Zirconium | Nobium | Molybdenum |   |   |   | Palladium | Silver | Cadmium |   |   |   |   | Iodine | Xenon |
Cesium | Barium |   | Tantalum | Tungsten |   | Osmium | Iridium | Platinum | Gold | Mercury |   | Lead |   |
I | II | IV | V | VI | VII | VIII | IX | X | XI | XII | XIII | XIV | XV | XVI | XVII | XVIII |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
H |   | He | ||||||||||||||
Li | Be |   | B | C | N | O | Fl | Ne | ||||||||
Na | Mg |   | Al | Si | P | S | Cl | Ar | ||||||||
K | Ca | Ti | V | Cr | Mn | Fe | Co | Ni | Cu | Zn | Ga | Ge | As | Se | Br | Kr |
  | Sr | Zr | No | Mo |   |   |   | Pd | Ag | Cd |   |   |   |   | I | Xe |
Cs | Ba |   | Ta | W |   | Os | Ir | Pt | Au | Hg |   | Pb |   |
The important thing here is that the elements in each column have the same chemical activity.
If Silicon is a semiconductor, then germanium and Carbon are as well.
Some of these elements are more important than others. Most of the world's industry is built around a handful of elements. Oxygen, nitrogen, hydrogen, sulfur, carbon, silicon, iron, copper, zinc, tin, lead, and aluminum would be a good list.
These elements are used in so many ways that a detailed list would fill a library. The use of metals is obvious. Pure silicon is used in making electronic devices of all sorts, and silicon dioxide is the main component of glass. The rest of the elements on the list are used to make millions of compounds with all sorts of uses.