Learn Polish: Nie widzę go dzisiaj. - I do not see him today.

In Polish, the verb ending usually tells you who is doing the action. Widzę specifically means "I see", so you don't need to add ja unless you want to emphasize it (for example, if you wanted to say "Well, I don't see him").

I learned that adding "nie" changes the object to the genitive case. Why are we still using "go"?

You are completely right about the rule! A negated verb requires the genitive case instead of the accusative. However, for the pronoun on (he), the accusative and genitive forms are exactly the same: go. So the word is in the genitive case here, it just looks identical to the accusative form you already know.

Could I put "go" at the very beginning of the sentence, like "Go nie widzę dzisiaj"?

No, go is a "short" pronoun, and a general rule in Polish is that short pronouns cannot start a sentence. They prefer to tuck in right after the verb or another stressed word, which is why it sits perfectly after widzę here.

Why does "nie" come right at the start? Can I separate it from the verb?

When you want to say you are not doing an action, nie must go directly in front of the verb it negates. You cannot put other words between nie and widzę.