The StarShip is moving relative to "Real Space".
You cannot, by definition, measure velocity relative to "space," real or otherwise. "Space" is just the set of coordinates you are using to judge the distance to other objects within your frame of reference. If those other objects are not moving towards you, or if there are no other objects to measure distance, then your relative velocity is exactly
zero.
Think about this analogy: pretend that you, the person is the StarShip and are about to go to Warp.
You're standing on the open cargo bed of a Pickup Truck and the Pickup Truck is your Warp Engine.
When the Pickup Truck moves, you're analogous to going to Warp.
You need to "Brace Yourself" so you don't move relative to your position within the Truck Bed.
Yes, that's newtonian motion. Because the pickup truck is being moved by the force of its wheels being rotated by the engine. The wheels apply force to the road (action) and are caused to move forward (reaction). The force is transmitted from the wheels to the frame, from the frame to the body, and from the body to you. If you are not properly braced for motion, the force will only be applied to the soles of your feet, which will move out from under you and you will fall over. If you are sitting down, the force is transmitted through the chair to your back and you are moved forward along with the truck.
If the truck accelerates really really fast, then you will crash into the back of the truck really hard, because the entire vehicle is accelerating forward and you, who are not attached to the wheels, are not. You only begin to move because the truck -- whatever part of it you're attached to -- is moving too.
Now picture this same pickup truck going over the side of a cliff and beginning to fall. If you take wind resistance out of the equation, what's going to happen? Of course, the truck is going to begin to accelerate towards the ground at 9.8 meters per second. Are you going to be propelled upwards into the top of the truck and pinned the roof of the truck as it accelerates downward? No, you are not. Because gravity attracts ALL objects at exactly the same acceleration -- 9.8 meters per second -- in which case it doesn't matter whether you're attached to the truck or not, you will accelerate at the exact same speed.
This is because a gravitational field is a non-inertial reference frame: the motions of bodies within a gravitational field DO NOT strictly follow newton's laws of motion, and are in a condition such that they will continually be in motion in a particular way whenever they are within this frame. This is why astronauts experience weightlessness in orbit, and -- by the same token -- why
people in an aircraft accelerating straight towards the ground at 1G also experience weightlessness.
So no, inertial dampeners are not needed to accelerate to warp speed, because the warp field will propel everything IN that field at a uniform acceleration.
More likely, inertial dampeners are needed to keep the crew from being thrown against the bulkheads whenever the ship
turns. For a ship the size of the Enterprise-A performing a 15 degree per second turn, people on the forward edge of the saucer section would be thrown towards the bow of the ship at a little over 1G. A ship the size of the Enterprise-D making the same turn (15 degrees per second is fast enough to turn completely around in 12 seconds, which the -D is more than capable of), then everyone in Ten Forward would be splattered against the windows at 2.2Gs. Inertial dampers would DEFINITELY be useful in that scenario, and the tollerances would be a lot lower (they could miss up to 10% of the imparted force and the worst that could happen is somebody spilling a drink).