Let's take the Alpha Centauri system as an example. The Alpha Centauri system is a system of three stars, the closest of which is Proxima Centauri, orbiting about 4.24 light years away. What's especially interesting about Proxima Centauri is that it has an Earth-like planet, apparently within its habitable zone, which might be a good candidate for human colonization outside of our own solar system. But the 4.24-light-year distance means that even traveling at the speed of light, it would take over 4 years to get there. Traveling at ten percent the speed of light, it would take ten times that—over 42 years. Traveling at only one percent the speed of light, it would take 424 years to get there. Assuming we want to be able to make the trip in a single lifetime, it looks like ten percent the speed of light is the bare minimum speed we would need to be able to achieve.
To put all of this in perspective, Voyager 1, currently the fastest spaceship in existence, is traveling at about 17 kilometers per second. That's about 0.00567% the speed of light (half of one ten-thousandth the speed of light). Of course, Voyager didn't start off at that speed—it only got that fast after multiple gravity assists from the giant planets—but for the sake of this discussion, we'll ignore that. At its current speed, if Voyager were pointed in the right direction, it would take 74,780 years to get to Proxima Centauri. That's a long time. But let's say that we somehow managed to double our top spacecraft speed every decade. It would still take us 80 years to get to one percent the speed of light, meaning that it would only take 424 years to get to Proxima Centauri. (Actually, after 80 years, we'd be at 1.45% the speed of light, but at that speed it would still take 292 years to get there.)
Assuming we could continue the trend of doubling our top speed every decade, after 110 years, we would be capable of traveling at 11.6% the speed of light, meaning that we could get to Proxima Centauri in only 36 and-a-half years. This means that a spacecraft sent 80 years from now would actually get there after a spacecraft sent 110 years from now: 80 years + 292 years to get there = 372 years, compared to 110 years + 36.5 years to get there = 146.5 years. And this brings us to what I will call the space-farer's paradox: The longer we wait to send a spaceship, the less time it will take to get there.
The space-farer's paradox means that it may not be worth sending a spacecraft until we can arrive at a point where the technological gain from waiting no longer outweighs the temporal gain from sending a spacecraft earlier. If we continue our thought exercise of doubling our top speed every ten years, this balance is reached somewhere between 140 and 150 years from now. After 140 years, we would have arrived at the ability to send a spacecraft traveling at around 93% the speed of light, which means it would arrive at Proxima Centauri after a voyage of only 4.6 years. Shortly after 140 years we would arrive at the ability to send a spacecraft traveling at the speed of light itself, beyond which any further progress is impossible, based on our current understanding of the universe. Even if it were possible, waiting another decade to double our speed from 93% the speed of light to 186% the speed of light would yield minimal returns, since the decade it would take to develop the technology would take longer than the difference in travel time—a spacecraft sent 140 from now at 93% the speed of light would arrive 145 years from now, while a spacecraft sent 150 years from now at 186% the speed of light would arrive 152 years from now.
Of course, this is all a little too hypothetical; doubling our speed every ten years is probably far too ambitious an assumption. Instead, let's use actual data to predict how fast our spacecrafts' speeds will increase. Voyager 1 launched in 1977, with a launch speed of 38,030 kilometers per hour. The New Horizons spacecraft that flew by Pluto in 2015 launched in 2006, with a launch speed of 58,536 kilometers per hour. Although their actual speeds at different points in their missions varied, the difference in launch speed between Voyager 1 and New Horizons gives us an idea of the progress we have made. This difference equates to an increase of speed of 154% over 29 years, an average about a 5.3% speed increase each year. With this more modest—and far more realistic—speed increase, we would arrive at 1% the speed of light 120 years from now, and would pass 10% the speed of light around 170 years from now. The benefit of waiting actually tapers off around 180 years from now, at which point we would be traveling at 18.5% the speed of light. At this speed, it would take only 23 years to travel to Proxima Centauri, meaning that a spacecraft launched 180 years from now (in 2197) could arrive as early as 2220. Once again, waiting to launch is no longer worth it after this point, since a spacecraft sent ten years later would only get there eight years faster, meaning that it would actually arrive two years later.
While 23 years might still seem like a long trip, it's a lot shorter than the nearly eighty thousand years it would take at current speeds. A 23-year voyage would mean that young astronauts could be sent to Proxima Centauri and arrive before turning 50. (At these speeds, the effect of time dilation is minimal; the 23 years as observed from Earth would equal about 22.9999608 years from the astronauts' perspective—a difference of only about 5.5 hours in a span of over two decades!) If colonization were the goal, young people could even be sent and arrive while still within child-bearing years.
Or course, we've only been talking about how long it would take to get to Proxima Centauri. Unfortunately, this doesn't take into account the difficulty of slowing down once we get there. Because of this, the first missions to Proxima Centauri would necessarily be fly-by missions (like the New Horizons mission to Pluto), without the ability to stop once we get there. Only later would we develop the ability to slow the ship down enough to orbit a target within the system, and only after that would we be able to develop the technology to land there. But those are discussions for another time.
