Should I edit resolv.conf in my machine, or leave it 192.168.1.x and change something in the router? If in the router, how?
If you need to modify the nameserver addresses (because, e.g., your router's nameserver is slow) you can set them in the NetworkManager's Connection Editor.
If you don't use NetworkManager then, in Ubuntu 11.10 or earlier, add a line to
In Ubuntu 12.04 or later you cannot edit
Since router IS the device that is connected to internet, you should play with DNS in router's GUI. How to get into router's interface and where to edit DNS, depends on what router you're using but i'm sure googling something like
will help. And yes, since it's router, it doesn't matter what OS you're using.