How does STAB work exactly, and how does it interact with the Technician ability?

The Power of an attack directly correlates to how much damage you deal, because its presence in the damage formula is entirely multiplicative. So, an attack with 120 power will do twice as much damage as one that has 60 power. The same goes with your corresponding attack stat: a Pokemon with 100 Attack will do twice as much damage as one with 50 Attack when using the same attack on the same target.

STAB has operated consistently since Gen 1 so this applies to all of the games. It increases the Power of an attack by 50% if it matches one of your types. 60 becomes 90, for instance.

Technician increases the Base Power of an attack by 50% if its Power is 60 or less. So the most you get is boosting an attack at 60 power to 90. This occurs before STAB is calculated, so your Power would be 90 for STAB, boosting it to 135.

The formula for attack damage can be found from bulbapedia. The formula for it is:

Damage = ( (2 x level + 10 / 250) x (attack/defense) x base + 2) x modifier

The part you're interested in is the modifier, it is calculated as:

modifier = STAB x Type x Critical X other X random[0.85,1]

In your case the the technician makes the other variable 1.5 when attack > 60. And when getting a STAB bonus, the stab variable is replaced by 1.5. So in your case STAB and technician should stack according to the information found from bulbapedia.