If, in vacuum, light is emitted from a source, would it accelerate really really fast until it reaches the well accepted speed of light $c$? If so, how small an interval does it take for it to do so?
Light doesn’t accelerate in vacuum. It only moves at one speed, $c$.
This can be understood classically as a consequence of Maxwell’s equations: when one derives the equation for electromagnetic waves from them, there is one unique speed at which the waves propagate.
Alternately, it can be understood based on relativistic quantum theory: massless photons can only move at the speed of light, or else they can’t carry energy and momentum.