Train Amsterdam to The Hague

Prices do not really vary. Tap is just the normal price, if you buy a ticket from the machine or a window, it will be a euro or two more. For tap, you each need a card, unless your grandchild is under 12, then they travel free. The same card would work if one has the physical card and the other has it in Apple/Google wallet, or both have it on your phones.

If you have a phone with service, you can buy from the NS app as well, the app is also handy for finding train options.

Prices for train tickets don’t increase nearer to the travel date. Tickets bought online via the official website of NS or in the NS app cost the same as tapping in and out with your credit card. Tickets bought at a machine or ticket counter cost €1 more. This is to cover the cost of the disposable chip card the ticket is printed on.
For all details about the tapping in and out, please see the website https://www.ovpay.nl/en

One important addition to the post by Paul. Please be very mindful that children aged 4 up to and including 11 only travel for free if they have a personalized OV Chipcard with their photo and details on it and the special Kids Vrij (free) subscription on it. This personalized OV-card costs €7,50 and you need to apply for it online. This is therefore not really practical for tourists or other occasional train users. In these cases the special Railrunner ticket is useful. For only €2,50, kids aged 4-11 can travel by train for one entire day. See here for details https://www.ns.nl/en/tickets/railrunner
So if you travel by train with a child aged 4-11, you need either a personalized OV card with a Kids Vrij subscription or a Railrunner ticket. This is different from other countries where kids can travel for free as long as you have something to prove their age.

If your only excursion during your visit is from Amsterdam to The Hague, there is a “ Tourist Day Ticket” that covers buses, trams and metros in Holland. This would cover the cost of taking a tram or metro from your hotel to the bus station in Amsterdam, the bus ride to The Hague and then the tram or metro best for reaching the art museum you are visiting. It also would have you covered for the return trip.
The cost of the ticket is just €15.50 per person.
Details can be found at www.TouristDayTicket.com

The €15,50 tourist day ticket linked to by Kenko is only valid on public transport in the province of South Holland. It’s not valid elsewhere and therefore not valid throughout the entire country.
Amsterdam is in the province of North Holland, so this ticket can’t be used for public transport within Amsterdam nor can it be used on the train from Amsterdam to The Hague.

Thank you, everyone!

I have downloaded the app. And I bought my tickets. It cost less and seemed simpler to get the task out of the way.

Looking forward to showing my granddaughter Amsterdam for her 16th birthday.

Wishing you a great adventure with your Granddaughter!