To control motor speed you need to pulse the power to the motor with a variable width pulse. The PICAXE has a very convenient command for this, pwmout. It runs in the background once executed, so will permit you to execute other code whilst continuing to output the pulses.
I'm not familiar with the particular board, but I assume that it uses an L293D dual bridge. In that case the pulses from the PICAXE can be fed into the two enable lines (legs 1 and 9).
The command pwmout Pin,255,N will give a variable pulse duty cycle (on to off time), with N=0 stopping the motor, N=1023 giving full speed, and anything in between giving part speed. "Pin" is the PICAXE output pin connected to the L293D enable pins, so:
pwmout 2,255,512 will give half speed (roughly) with PICAXE pin 2 (leg 5) connected to the L293D enable pins.
The frequency of the pulses will be about 3.9 kHz. Other (higher) frequencies are possible by using different values in the pwmout command (try the PWM Wizard in the Program Editor).
Hope this helps.