I'm no networking guru, but you don't have to be one to do this either.
Let's say your current router's IP (and usually this is your default gateway) is 192.168.1.1.
On the new router, plug it directly into a PC. By default it should have its DHCP server enabled. Do an IPCONFIG/ALL to get the default gateway. Go to this IP and log on using your web browser.
The first thing you want to do is turn OFF the DHCP function. Your PC should keep its address but if it doesn't that's ok as you can assign one temporarily in order to connect to the router.
Log back in the router and change its IP to 192.168.1.2 (it's normal for the DHCP client range to start at 100 and if not on the first router change it so you don't have an overlap!)
Set up your SSID and security for your second router WLAN as you wish. If your SSID on router 1 is called mywirelessnetwork you may want to name the second in a similar fashion; such as mywirelessnetwork1 etc.
Finally when connecting your second router to your network just plug in one of the switch ports to your existing network. Do NOT connect the WAN port to anything. The second router will function as an access point like this and give you better coverage as you need.