The speed of light is not a speed limit on effective travel, but on observation. Imagine you build an engine that gives enough thrust to roughly produce an extra effective 1km/s of velocity each time you operate it. Even though the speed of light is 300km/s, that engine is going to operate exactly the same on the 301st operation as it did on the first operation. And, in fact, in 'real' terms if we create an engine capable of generating 1g of thrust for years at a time - we could travel 700 light years in ~12 years by thrusting for 1g forward 6 years, turning around and then thrusting for 6 years backwards.

But of course the 'catch' is that time dilation (and length contraction) kick in. And so while from the frame of reference of the ship and all its travelers only 12 years will have passed, from the frame of reference of everybody back on Earth - 700 years would have passed. What we would observe is that as the ship approached the speed of light, its apparent mass would approach infinity, reaching a velocity asymptote at the speed of light.

And these values are also not linear. It's possible to travel a practically unlimited distance in the span of a single human lifetime - from the perspective of that human. 100 trillion light years would take about 62 years at 1g from the perspective of those on board our ship. There's also really really fun paradoxes available here. Imagine somebody leaves Earth towards this planet and at some point decades, or perhaps even centuries later, we develop technology that enables substantially faster travel. Our planetary settlers on their 12 year journey could arrive 12 years later to find that not only is the planet indeed colonized, but it's colonized by the descendents of Earthlings that left decades, or even centuries, after they did.

The really exciting thing here is that NASA seems to have gone silent on the EM drive after having it pass every single test they threw at it. In my opinion, this is evidence of classification which is something that would make sense. If the EM drive somehow actually works - and everything we know says it most certainly should not - that would change absolutely everything. The ability to actually start realistically considering these sort of missions would be mundane compared to other possibilities, which I'm not even going to get into as they sound, and should be, absurd. As an aside, this part on the EM drive is 100% speculation - not to be confused with everything stated above, which are direct and heavily tested (even if absurd sounding) consequences of relativity.