how fast are microcontrollers?

[GiTS]

Could I obtain data from a sensor, process an equation and a few lines of code, and output a signal to something like a speaker in .001 seconds or less?

[mgb_phys]

Depends on the uC and the amount of calculation , but generally yes.

Take a look at this for real-time audio on a popular uC
http://interface.khm.de/index.php/labor/experimente/arduino-realtime-audio-processing/

[GiTS]

what's "uC"?

what would tell me when I look at the specs for a device, how long it takes to transmit and recieve data?
for instance, this accelerometer http://www.sparkfun.com/datasheets/Sensors/Accelerometer/ADXL103_203.pdf

[Maxwell]

As I am sure you know, that is not a microcontroller (uC - u for micro, C for Controller) - it's an accelerometer.

As for it's output ability:

The user selects the bandwidth of the accelerometer using Capacitor CX and Capacitor CY at the XOUT and YOUT pins. Bandwidths of 0.5 Hz to 2.5 kHz may be selected to suit the application.