Starting with the sugar, why is that we can make a beer or wine. What is the yeast doing with the sugars? What is gained? What is lost?

So far I have the sugar interacts with yeast and creates carbon dioxide and ethyl alcohol but other than that I am struggling.

Might want to verify this as I'm much more of an inorganic chemist than a biochemist. Sugar is broken down in glycolysis into pyruvate. In our bodies this is done with oxygen and it is called a aerobic reaction. We are further able to break down pyruvate all the way. As a result we breath out carbon dioxide.

Yeast do a smiler process without oxygen, thus anaerobic reaction. The pyruvate is broken down to acetylaldehyde and then reduced to ethanol.

Side note: Humans take ethanol and convert it to acetylaldehyde when we drink. However if methanol is consumed then it turns to formaldehyde which is very toxic (and we use it to embalm people). This is what makes you blind from moonshine.