OK, I've managed to fix it so I can use the internet through a browser. However whenever I try to download something using commands such as:
it wont work. I instead have to first download the file using a browser and then install it using the rpm command.

At the moment I am trying to do a yum update, but keep getting the response of "No more mirrors". I have a feeling this is related to the original problem.

The only possible reason I've managed to come up with to these problems is that my DNS isn't set up correctly. Currently the only nameserver in my /etc/resolv.conf file is 192.168.1.1. I've tried to add the DNS values which are in my router settings but the file is reset back to its original state if the computer is restarted. Is there a way to stop this happening?

If there is any other reason to why I can't download things through these commands please let me know, as I have searched quite a bit and haven't come up with anything.

I don't believe rpm will retrieve files then install them from the internet. You will need to download them first to install them that way. Check your yum config file and look thru it. maybe the mirrors are old or something. Yum should use http or ftp to download and install files, the same as your browser, so if your browser connects to the internet, so can yum.