This is the first I knew that ASUS made switches, though I cannot say I am surprised.
First, ensure your router is handling the NAT, DHCP, and DNS services for you.
NAT is the address translation form the IP address your ISP gives your circuit to a private network such as 10.x.x.x or 192.168.x . Each machine will need its own IP address which will be handled by the DHCP service built into most routers these days. The DNS service is also built into most routers as well. It serves to translate the name you type in to an IP address so that your destination can be found on the Internet. You should find the information about this in the documentation that came with the router. If you do not have documentation, call your ISP for help with it to ensure it is setup properly.
Get some CAT5e cables. They should be the length you need to attach the router to the switch and each computer with the switch. Connect the router to port 8 unless the switch has a specific port labeled for Internet or WAN. When the cable is connected, and the circuit is active, a light near the port or labeled with the port number should light. If it does not, then there is a problem. Likely it would be with the cable. You may have gotten the very rare bad cable, or the connection requires a cable known as a cross-over to be used–this is also pretty rare these days. The port could also be bad. Try plugging the cable into another port, if it works then, you probably should return the switch for a replacement.
The next step is to connect your first machine. [I am assuming all machines have a network connector built in or previously added.] Check to see that you can get out to the internet. If you are successful, you now have one computer connected. If you are going to want to share data and/or devices between computers, check and not the Workgroup name–it will need to be the same on all machines in order to be able to share.
Proceed to the second and third machines. Change the workgroup name if needed. test. Share something to ensure the machines are actually seeing each other. Have fun..