Amsterdam to Haarlem

Your first answer is the correct one.

Or you can take a bus direct from the airport to Haarlem. The best answer will depend on where in Haarlem you want to go.

Do we take a train from the airport to the main train station in
Amsterdam

No. You would take the train to Amsterdam Sloterdijk station which is one stop before Amsterdam Centraal, and from there transfer to a train to Haarlem.

You can check rail routings via the Dutch Rail site:
https://tinyurl.com/y9nad73x

Also be sure to install the official Dutch Rail app on your smartphone. It's very easy to use and you can even purchase tickets right on your phone.

You can use both stations - Sloterdijk and Centraal - to change trains. Sloterdijk can be confusing, being on two levels. Centraal is straightforward.

In April we took the bus
"for a more direct route (cheaper and more convenient although it takes about 10 minutes longer), take Bus 300 called the Zuidtangent which runs every 10-15 minutes between Schiphol airport and Haarlem Central Station. The trip takes about 30-40 minutes. Buses are signposted when you exit Schiphol airport, and the Zuidtangent is a big red bus with white writing on a grey background – look out for the number 300 on the front."
This was easy and an interesting trip seeing the area for the first time. No transferring.

Are they still using the bendy bus on the 300 route? They used to ....