Generally, about how long after a flight takes off can it be tracked on FlightAware?
Departures are not from the gate but start at ’ wheels up '. Add the six minute delay and you are looking at least ten minutes or so from the time the flight leaves the gate to when you see it on FA.
Many airlines on their web pages will inform you when a flight has departed its’ gate and thus is taxiing into position for take off.
If you do some searching you may find some varied answers on this.