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There are two types of people in this world. The first (type) is those who do that, and the second (type) is those who don't.
There are two types of people in this world. The first are those who do that, and the second are those who don't.
Which is the right verb to use here? Are they both fine, with is putting an emphasis on the type and are the people?
Both are fine and natural.
The noun phrase "type of people" can be understood as a single unit, the type, or the plural group that it represents.
Grammatically, it's singular, so it can be treated as such.
But in that sentence, the subject complement, "those who..." is plural, so it clearly can be thought of as plural as well.
There are many group nouns like this that can be treated grammatically as either singular or plural, including "staff", "family", "team" and so on.
It should be is. This is because you are talking about types. It's helpful to put the thing you are talking about after the first just as you have done. If you were to say "The first type are those who do...", it wouldn't be right.
