I am not talking about going to some sites and test your internet speed, but rather an in-house speedometer, so to speak. In Ubuntu (or Gnome) it's easy, just install Netspeed applet in your Gnome Panel.

Right click your panel > Add to Panel > Network Monitor, like this :

And you'll have something like this :

0 B/s, I know, sad.

Right click the applet to customize it appearance.

If netspeed is not listed already in your applets list, you have to install it via Synaptic Package Manager, "netspeed".