From zero to 1000c+ takes a few seconds, fly far away from the star until your speed is 1000c+, go to zero throttle, then set max throttle, you will see how fast it is. SC speed is limited, as already stated, by the local gravitational field not by any imagined rate of acel/decel. If you fly towards a star at max throttle you will slow down, if you fly away from a star you will speed up, if you fly in a circle around a star maintaining the same distance you will neither speed up or slow down, this is how SC works. The best way to get up to speed is to fly directly away from the nearest local source of gravity. Understanding how SC works with gravity will help you get around faster.
For instance if you want to scan moons on opposites sides of a gas giant the longest way to do so is to fly directly from one to the other because you are flying into the gravity well of the gas giant and then out again, you loop around keeping distance and you will be there in a 10th of the time. There is virtually no decel/acel in SC, it's almost instant, the moment you hit max throttle you are at the max allowed speed for the local conditions, that's basically as fast as you can go. Consider it like a car speed limiter tuned to GPS that governs your speed according to local speed limit laws, except in this case it's governed by the laws of imagined physics and not legislation. if you put your foot down in a powerful car you can get to say 60kph in seconds, but it doesn't matter how hard you press, you won't go any faster until you leave the 60 kph speed zone.
If you are taking a long time to get to 1c it is because you are flying through a slowly decreasing gravity field at an angle, fly directly away from the source of gravity to get to 1c faster. The rules of SC are consistent across our imagined galaxy and universe, messing with them would be a bad idea, for instance if you just changed the way SC worked so you got to 1c much faster every time you entered a new system you would face plant into the star, making zero throttle mandatory for system entry and slowing everyone down, and that wouldn't go down well.