We're supposed to think that full impulse is a viable interplanetary speed, representing a significant fraction of lightspeed - yet acceleration to full impulse, when depicted in exterior visuals showing a point of reference (say, a fellow ship), tends to be very gradual. Is that just a matter of caution, though? We do have evidence of rather explosive accelerations, such as the one where our heroes leave Earth in ST:TMP - perhaps in a hurry, certain safety considerations are forgotten?
In TOS, we've seen a full power maximum impulse acceleration away from a planet in "The Squire of Gothos" and it was a crazy explosive acceleration. When they left orbit in "Tomorrow is Yesterday" the acceleration is comparable to TMP. I think on average, it will be gradual. But if Kirk orders Scotty to throw all power into it, then it is pretty immediate.
That seems also to be the case with warp acceleration. If all power is thrown into it then the acceleration is almost immediate. But if they can only feed the engines a fraction of the power as the rest is taken up by shields and weapons or if the engines are just starting up from "idle" then the acceleration would be gradual, IMHO.
As to TNG - they might follow that same trend but I'm not positive.