That should be very easy.
When the mini is booted up, you'll see the Airport icon in the menu bar - it looks like a white fan shape if airport is off, and a set of up to four black arcs (the more there are the stronger the signal) if it's on.
Just click on the icon and a menu will drop down listing all the visible wireless networks. If yours hasn't been given a name, it'll probably appear as 'linksys'. If your network is showing, select it and if you have no security enabled, the mini will join it. If you have WPA or WEP enabled, you'll be asked for a password. Enter the password you've set for your router, and you'll be in.
It really is that simple - usually! If you have problems, just post the details of any errors or a description of what happens, and we should be able to get you running pretty fast!