There is a difference between a "wireless router" and a "wireless access point". Since you already have a router, you only need a wireless access point, the one from Linksys is a snap to install. You could use a wireless router , but installing it will be more involved because you have to disable all the crap you don't need and it might conflict with your existing router, giving you a PIA. Go buy yourself one of these:
http://www.linksysbycisco.com/US/en/products/WAP54G
Or a better model, there are a bunch of them out there, just make sure it is an "access point".
When you run the install, make sure you select WEP key security, it's the easiest. If you don't secure it, the whole neighborhood will be using it. Some hacker crooks just cruise around neighborhoods looking for open wireless connections.