Chemical symbols are probably the first thing you notice when you look at the periodic table. These are the one- or two-letter abbreviations for the elements themselves. It's a little imprecise to call them "chemical" symbols, since elemental symbols would be more appropriate, but that's what they're called. Some of them are pretty intuitive, like "H" for hydrogen, but some are less so, like "Au" for gold and "Ag" for silver. In these cases, it's a clue that the element was known in the "ancient" world, i.e. Rome, and received a name, typically a Latin one, prior to its classification as an element according to modern chemistry. Gold, for example, is "aurum" and silver is "argentum" (coincidentally, this is where Argentina got its name).
A chemical formula is the combination of elements necessary to form a particular molecule. The one most familiar to many people is H2O, indicating that the molecule is made of two hydrogen atoms and one oxygen atom. Another would be CO2, which is composed of one carbon and two oxygens. Think of it like an "ingredient list" for making the compound. Note that it doesn't tell you what the actual arrangement of the molecule is and, for more complex compounds, the chemical formula alone won't be enough to specify exactly which molecule you're talking about.
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