Learn Russian: Во-первых, я не знаю этот город.

The prefix во- and the suffix -ых are added to ordinal numbers to create list markers. It literally translates to something like in the firsts. To say secondly, you take the word for second (второй) and apply the exact same pattern to make во-вторых.

Because there is a negation with не, shouldn't этот город be in the Genitive case?

You can sometimes use the Genitive case for a direct object after a negated verb, which is known as the Genitive of negation. However, modern Russian strongly prefers keeping the Accusative case for concrete, specific, and tangible objects like этот город. Using the Genitive here (этого города) is grammatically possible but sounds much less natural than it would for an abstract concept.

Why doesn't the ending of этот город change here since it is the direct object of the verb?

It is actually in the Accusative case, but город is an inanimate masculine noun. For masculine nouns in Russian, only animate objects like people and animals take a modified Accusative ending. Inanimate masculine nouns look exactly like their Nominative dictionary form.

Could I use знаком instead of знаю to say I am not familiar with this city?

Yes, you could say я не знаком с этим городом, which literally means I am not acquainted with this city. However, using знаю directly with a place is perfectly natural and is the standard, everyday way to express that you do not know your way around a location.