Is there any way for me to configure my home router such that all outgoing requests coming from some of my local machines are rejected (deny them service), or better yet, redirect them to some other page?
Typically it is possible to restrict internet to certain computers on your network, however it comes with some caveats.
The $50 netgear router you picked up from Best Buy will likely be very limited on the restrictions that can be placed on a computer inside the network.
On my dd-wrt router, I am able to restrict computers by day, website, or protocol.
Another popular solution is to use a full fleged firewall distribution running on an old computer you have laying around.
There are many free software solutions.
These will be your best bet because they will allow you to have full control over your network, while still providing relatively simple user interfaces.
Lastly, you could opt for enterprise level filtering by using a captive portal.