Other people have already said this, but they're buried in replies to replies so I'll say this up here where it's more noticeable:
The practical upshot that a human can get to anywhere in the universe within their lifetime given enough fuel to keep up acceleration is correct, but from no frame of reference will you appear to have travelled faster than a beam of light.
In your traveling frame of reference, it will appear that the distance you travelled got smaller. That's why you can reach places that seemed too distant to reach in your lifetime before: because they don't seem so distant once you're on your way there.
In the rest of the universe's frame of reference, it will appear that you aged more slowly. That's why you can reach places that seemed too distant to reach in your lifetime before: because your lifetime got prolonged once you were on your way there.
In either frame of reference, when you get where you're going, you will still find that a beam of light sent at the same moment you departed will have arrived at your destination before you, and thus in neither frame of reference did you outrun the light. You just either aged more slowly or travelled less distance, depending on whose frame of reference we're talking about.
In a photon's frame of reference, there is no distance between anything and no at all time elapses to travel it. Given enough fuel you can get arbitrarily close to that and so travel to arbitrary locations with arbitrarily little aging along the way, and so get anywhere in your lifetime. But light can always do that better than you still.