Do you mean a induction motor or series motor as found in drills, vaccum cleaners etc. Series motors have brushes and change there speed with load.
Series motors are easy to control manualy, plenty of drill speed circuits about, even light dimmers (probably becoming scarce in your part of the world) would work on a small motor.
You would need to figure out some way of replacing the pot with a signal from the picaxe. PWM->optocoupler->dimmer could work.
They rely on detecting the zero crossing point of the mains supply sine wave, so NO DIRECT CONNECTION. Look for subjects like "simple circuit using triacs". I have a little paperback called 101 things to do with a thyristor, it is a wealth of information.
Induction motors can be controlled by this method but over a very limited range. You need to alter the frequency. All the inverters I have used are 1ph input up to 1.1-2.2kW with 3ph output, and 3 ph input above this.
Trying to alter the speed of a single ph motor would incur lots of problems. Start and run capacitors would need to be different values at different speeds for a start.
What kW was you thinking of.