G'day DAL2728,
PROF (& NAV if appropriate) armed during taxi.
The PNF presses AUTOFLIGHT when the PF is satisfied that both engines are accelerating uniformly - usually <30kias, this engages the autothrottle and sets take-off power, either flex or ref as set in the FMS.
Once airborne and at or above 400' AAL, the next press of AUTOFLIGHT will engage the autopilot.
rhovsquared
In simple terms PROF is the vertical flight path management of the aircraft. For climb it is usually pitch for airspeed coupled with climb power. For descent, it is usually pitch for a 3 degree descent to the runway from cruise altitude, with slow down segments to allow for speed reduction from descent speed to 250KIAS < 10,000' and then reconfiguring for landing.