I SO disagree with a lot of the advice in this thread.
:thumbsup: I hate it when people just say shoot with a faster larger aperture as the end all solution. It's one way to obtain a faster shutter setting but not the only nor the best way.
To add to Nate's response....
You have two different exposures (at any given ISO) involved and each is controlled differently.
First, you have ambient exposure. This is controlled by the combination of Shutter and Aperture.
Second, you have exposure from flash. This is controlled by the flash's output and aperture. As Nate pointed out, shutter setting doesn't matter for exposure from flash (as long as you stay under sync speed).
If you adjust the aperture, you are changing the exposure of both ambient and flash
If you adjust the shutter, you are changing the exposure of just ambient
If you adjust the flash's power, you are changing the exposure of the subject illuminated by flash.
I recommend some good reading:
www.strobist.com for more details.
btw... I would rather shoot with an older flash in full manual than resort to using the on-board flash. Better yet... get the flash off of the camera. (warning be wary of trigger voltages when choosing older flashes).