Is there any easy way to track the route that an request takes in Ubuntu? For example, tell me what the DNS servers are, which servers it redirects through before reaching the final page, etc. Is there any program or command that I could do this with?
The Traceroute tab in "Network Tools" - hit the super key, enter "Network Tools", or enter gnome-nettool in a terminal.
On the whole, the Network Tools should fulfill most of your network-related needs - ping, finger, whois,...
For newer versions of Ubuntu (14.04 and beyond), install and user the package named 'traceroute'