Eventually everything on the same circuit most be connected to the same grounds. All of the ground pins you have on the Arduino are connected together, the multiple pins are provided as a convenience. I imagine for high voltage situations there are other factors, but in general you want to connect the grounds.
Now for servos, they likely will overload the power supply of the Arduino, so what you typically want to do is have two power supplies, one for the Arduino, and one for the servo. You do wire the grounds of the Arduino and servo power supplies together. Duane B wrote this helpful article on how to wire up servos: http://rcarduino.blogspot.com/2012/04/servo-problems-with-arduino-part-1.html