I think you are almost there.
Here is a checklist:
1. Set the IP of the Linksys to a static IP. This IP should be in the subnet of your (DHCP server) network, but not within its DHCP scope. Make certain it is not in use by another device.
2. Disable DHCP on the Linksys.
3. Change Gateway to Router mode on the DHCP Screen. You can use RIP-1 for both TX and TX.
4. Set you SSID, Channel and other settings on the Linksys. For the moment, leave WEP out of the equation until everything else is sorted.
4. Run a straight-through Cat5 drop cable from a normal port on the Cisco to the uplink port on the Linksys.
5. For the Wireless clients, set the SSID and Channel to values matching the Wireless router. Set them to Infrastructure mode. Disable 802.1x authentication if shown as an Advanced option.
Try a wired client on the router. It should be able to obtain a DHCP, and have access to the local LAN and the Internet.
Try a wireless client. It should now work.