If general relativity is an accurate model of gravity then distortions in space-time are caused by mass. Traveling at the speed of light can be shown to require an infinite amount of energy to accomplish for a finite mass. However space-time itself could, according to our current understanding, travel faster than the speed of light.

So if you could 'warp' space around you rather than travel through it, you could go at an arbitrary speed. Unfortunately warping space-time is typically accomplished by large amounts of mass. Hence a large amount of energy. If I remember correctly it also requires we bend space in a way we have never seen nor have any reason to believe is possible to accomplish beyond it being allowed by general relativity. It's quite a pickle.

Without such a device humanity will still populate the galaxy but it'll take hundreds of millions of years. Assuming we don't wipe ourselves out.