Information about the metro can be found here:
https://www.nexus.org.uk/metro
You have to buy a paper ticket to travel on the metro. You can use credit cards to buy tickets, but you can't use a credit card as a ticket.
There is a smart card alternative to paper tickets, which is cheaper, but you cannot buy it at the airport. You can buy it online and have it sent to a UK address.
The most convenient hotel is the Hampton by Newcastle Central station. All shopping you could want to do is accessible from that hotel on foot.
There is a Doubletree by the airport which may be useful for a short stay, but repeatedly travelling to the city centre and back will get annoying.
There is a Hilton in Gateshead. It is a bit of a walk from the metro but you could just walk across the bridge to the city.
Sounds like Hilton is the better chain to stay in Newcastle. You think I can order that smart card and have it mailed to my hotel? Have you stayed at that Hampton before? Good value for the money?
TIA!