Actually, a wireless router include the following component:
1. a router that share ONE connection and NAT to all other computers.
2. a switch to connect computers
3. Wireless Access Point (AP)
4. DHCP server to assign IP to internal computers
Usually, there is 1 port for the internet (#1), 4 ports (#2) for internal network, and antanna (#3) for the Wireless AP. You can't see the DHCP server externally because it is just a piece of software.
To connect a wireless router to an existing working router, the best is to disable the #1 and #4 and only use the #2 and #3. It will prevent ping skyrocket problem said by osanowo. Also, it will not spilt the network as Mauk said. The file and printer sharing should work if you set up correctly.
Follow the steps below:
1. write down you existing router IP address and the DHCP range. e.g Router IP is 192.168.1.1 and the range is 192.168.1.100 to 192.168.1.200
2. Connect one of your computer to the internal network pot (the 4 ports) of the new wireless router. Open the router set up page and set the IP of the new Wireless Router to a value that is never used and outside the DHCP range. e.g. 192.168.1.230. please note, the first 3 set number (e.g. 192.168.1) should be the same
3. Disable the DHCP server in the new Wireless router. Note: since the IP is change, you have to use the new IP to open the router setup page.
4. disconnect the new Wireless router from the computer. Connect the new wireless router and existing router using the internal network port (the 4 ports) in both routers.
5. Connect the computer to the new wireless router internal network port (the 4 ports). Test if the computer can get the IP from the old router DHCP server (open command prompt and type IPCONFIG to view the IP). Test if it can go to internet. If the computer cannot get the IP, you have to manual assign an IP (e.g. 192.168.1.10) in the TCP/IP of Network Property. Then open the set up page of the new Wireless router and verify all the setting. After finsh, set back the IP to "Obtain IP automatically".
6. Open the new Wireless router set up page to set up the wireless network (e.g. SSID, Channel, Security)
PS: you don't have to disable the #1 in the new Wireless router. Just set it to DHCP client for the WAN (Internet) set up.
Final:
Internet connect to the WAN port of the existing router. 1 out of 4 internal network port is used to connect both routers. So there is still 6 (3*2) port to connect 6 computers by wires. Wireless network is provided by the AP in the new router.
For suggestion of the wireless router. Since you are not using the #1 part of the router, any 802.11G 54Mbps router is good for you. You don't need the 108Mbps or pre-N router. Those router require a matched adapter (same manufacturer is better) in your notebook to use the extra feature. Save you money for a basic 802.11G 54Mbps router now.