Answer to your first two questions is the train. You can use the NS app to see what public transport options are available, including departures times, walking routes to and from the station/bus/metro/tram stop and which platform the trains/bus;tram/metro depart from.
More information on travel options in Amsterdam can be found here: https://www.iamsterdam.com/en/travel-stay/getting-around/public-transport-in-amsterdam. The trains are operated by the NS and the city buses, trams and metro by the municipal GVB. Regional busses are run by mostly Connexxion. Haarlem is not served by the GVB.
I am not sure the various passes listed in the Save money on public transport with Amsterdam travel tickets section at https://www.iamsterdam.com/en/travel-stay/getting-around/public-transport-in-amsterdam are economically of interest to you.
Ideally, it is tap in and tap out with all modes of public transportation. It would mean though you will need to have one credit card per person or set up devices so each of you have a payment method (or a mix of adults with individual credit cards and the children with these same cards linked on phones or smart watches). I would search this forum for tips and experiences.
Alternatively, you could buy individual day return ttrain ickets on the NS app (which then also allows you to benefit from off-peak discount) or the vending machines/counter at Sloterdijk train station in combination with a GVB (multi-) day pass for city trams, busses and metros. Maybe you can just take the tram from Sloterdijk to where you want to go (like the to the museum quarter) with no need to take a train. The NS or 9292 apps will show you the options available.