The Internet is a big thing. Which bits of it are you trying to connect to, and how? Opening a browser and surfing the web should be as transparent a process as it would be on a Mac or a Windows PC. The requests go through your Netgear router (which is a 'Linux box', anyway, of course) and get rerouted to your ISP's DNS servers. The fact that the trffic has come from a browser on a Linux PC makes no difference to that process.
Getting your email may involve you finding out the names of your incoming and outgoing mail servers, but your ISP should be able to tell you those.
As long as your router is set up correctly, there isn't really a difference to using a Linux PC, to any other type of machine.