Many thanks to Mr Guzman for helping me sort this out! Here's what got my redhat8 and linksys 4 port router working...

1. You should have your cable modem (or whatever) connected to the WAN port on your router
2. Your redhat machine connected to one of the LAN ports on the router.
3. su to root on your redhat box and give the following commands ($=prompt):

     $ /sbin/ifdown eth0
     $ /sbin/ifup eth0
   
     This will bring the 'eth0' interface down and then up again, forcing a DHCP (DynamicHostControlProtocol) request to the router instead of to the cable modem.  If this succeeds without any error messages, then...
   

4. Run /sbin/ifconfig once again and verify that you have an IP in the 192.168.1.x range for eth0.
5. Run /sbin/route and verify that http://192.168.1.1/ (go on and click it! why not?) is set as your gateway.

   (Note that if the DHCP client is properly configured on your redhat machine, all these values should be correct.)

6. Return to the routers setup http://192.168.1.1/ and make sure it is getting the proper address from the cable modem (this will probably be 66.215.200.something).

   If it is, then you should be able to fire up a browser and get on the net.

7. Test it by getting to the net!

And Mr Guzman was right! It worked like a charm!

This was my set up:

On my RedHat8? box, my cable modem is set up this way:

  bash-2.05b$ cat resolv.conf
  ; generated by /sbin/dhclient-script
  search charterpipeline.net
  nameserver 24.25.195.3
  nameserver 24.205.1.62
  nameserver 24.205.1.14
  nameserver 24.205.192.36
  nameserver 24.205.224.36

My Linksys router I.P. address will be:

  192.168.1.1

I need to get my eth0 info as well I imagine:

  bash-2.05b$ cat ifcfg-eth0
  DEVICE=eth0
  BOOTPROTO=dhcp
  ONBOOT=yes
  USERCTL=no
  PEERDNS=no
  TYPE=Ethernet