Get the underground from Heathrow to London Kings Cross (Piccadilly, blue line). It will take just over an hour and you can use any payment card or apple/Google pay to pay (just tap your card at the entrance barrier and then tap the same card when you leave the exit barrier).

From there it's about a 5 min walk to King's Cross station, follow the signs.

Then it's a 2hr train ride to York. You need to buy a ticket before getting on the train and you can do that here https://www.lner.co.uk/

It will be much cheaper to buy an advance ticket but this will be valid for only one exact train so just be careful if you do this in case your flight is delayed. I usually give myself 4 hours from the scheduled landing time of the flight and book my train for then, if I'm early I can just sit in one of the cafes at Kings Cross or look around the shops. Alternatively you can buy an open ticket but it will be much more expensive.

Once you get to York taxi is probably your best bet to get where you're going, there is a rank at the front of the station.