What type of cable is needed to connect the Apple TV box to a television? I am being asked by a resort I'll be bringing it to and I need a more complete description than "coaxial"

You would need an HDMI cable (not coax) to connect the Apple TV box to the TV.

For Apple TV (4th generation or earlier), use a 10 Gbps HDMI cable or better.

For Apple TV 4K (1st generation), use an 18 Gbps HDMI cable or better.

For Apple TV 4K (2nd generation), use a 48Gbps HDMI cable, if connected to a TV with HDMI 2.1 port.