Learn German: Wir kennen die Gegend nicht gut. - We do not know the area well.

In German, kennen is used for being familiar with a person, place, or thing (like knowing the area in this sentence). wissen is used for knowing facts or having information, usually followed by a question word or a sub-clause (like knowing why the area is nice).

Why does nicht come directly before gut?

When you want to negate a specific adverb or adjective, nicht goes right in front of it. Here, you are specifically negating gut (not well), rather than negating the entire sentence.

What case is die Gegend in this sentence?

It is in the accusative case because it is the direct object of the verb kennen. For feminine nouns like die Gegend, the definite article remains die in the accusative, looking exactly the same as it does in the nominative case.

Could I use das Gebiet instead of die Gegend here?

You could, but die Gegend is much more natural for casually talking about a neighborhood or local surroundings. das Gebiet sounds a bit more technical, like a specific territory, zone, or administrative district.