Do I need to subscribe to Apple TV in order to set up and use it?
Subscribing to Apple TV+ is completely optional. An Apple TV+ subscription is for very specific content (Apple Originals). Other content would not rely on this subscription (but could require other subscriptions).
I purchased an Apple TV for my son a couple years ago
What model was that? Identify your Apple TV model - Apple Support
and he has not been able to use it yet...
It should be easy. Set up your Apple TV - Apple Support
perhaps we had an older HDMI cable.
Older HDMI cables with maybe insufficient bandwidth would still work, but perhaps at lower specification.
I have purchased an HDMI 2.1 cable.
I hope you got a good deal on that, as an HDMI 2.0 cable (18 Gbps) would have been sufficient. An HDMI 2.1 cable (48 Gbps) offers more bandwidth capacity than any current Apple TV is able to use.