I just built up the same kit. It's very basic but it works. If you can spare the space, put the 4x'AA' battery holder on the top of the chassis along with the arduino board, otherwise you'll have to remove the top plate to change the batteries.
I noticed that when driving the Arduino with a nice strong wall adapter connected to the Arduino power jack, the motors ran quite well. However, when using the onboard battery pack supplied with the chassis, the whole contraption ran a bit more sluggishly. It's obvious that the motors + Arduino + motor shield is a bit too much for the standard batteries. This unit would definitely benefit from a LiPo battery in place of the stock arrangement.
Well ,TwelveBarBrews, this motor driver breakout board has an external voltage pin, where I will connect the included battery pack or a LiPo battery , as you suggested. Also ,the arduino will be powered from a 9v battery.