Do I connect the ethernet cable that runs from my modem to the router directly to one of the computers?
Yes. Make sure that you have at least the built-in Windows firewall and your antivirus software running on that computer.

How does this configuration recognize the change from wireless to hard wired?
If you have your TCP/IP settings set to automatic, there should not be a problem.

Do I need shut down the other 2 computers on the wireless network to do this?
You don't need to shut them down, but it couldn't hurt. They won't have wireless internet access while you are using the new configuration.