Let's get one thing clear to start with:
Your laptops don't communicate directly with your printer. It's only your router that talks directly to the printer. (Since you've got a new router, that's why your printer had to be told how to communicate with it). Then any computer that can also talk to your router can use it to communicate with the printer. (Obviously, those computers must have the printer's driver software on them but, since it worked before, the troublesome laptop clearly has).
So you've got one problem, not two. (i.e. if you can get your laptop to talk to the router it will be able to use both the internet and your printer straight away).
You've not told us which version of Windows the laptop has on it, so I'll guess at Windows 7 (although over versions of Windows might be fairly similar). Take your mouse down to the right hand corner of the screen and locate the icon for internet connections. Click to see whether your laptop thinks it's got a connection or not.
If there's no connection shown, look for your router's name in the list of available connections and click to connect using it. You'll need to enter the password you should have been given for it. You should also ensure that 'connect automatically' is checked, so that you won't have to go through the connection process every time the laptop is started.
If there is a connection shown, but you simply can't use it, try resetting the laptop's TCP/IP stack. (It usually fixes such problems):
http://support.microsoft.com/kb/299357