Long story short: to travel to Alpha Centauri it would take for the traveller still about 4 years, and when back at home, people would have aged only some months more than the astronauts.
Crazy things happen if you decide to go on galaxy-wide trips though, to the astronauts because space contraction they perceive their own trip as happening still rather quick, a 100 000 ly trip (the size of our galaxy, for reference) takes 22 years for someone inside a spaceship with 1g acceleration, perfectly doable. But people outside still see it taking 100 000 years... (meaning they will be 100 000 years "older" from the point of view of the astronauts)
After I found all this out, I concluded space travel isn't THAT hard, assuming you have a way to accelerate constantly at 1g for 22 years (that is the hardest part actually), you can get anywhere in the galaxy in a human lifetime, no need for generation ships, cryogenics or other crazy tech.
In fact even going to other galaxies is easy, a trip to Andromeda takes 28 years!
Mind you, all those calculations were done assuming you will burn at 1g until half the distance, and then burn at 1g to brake, if you don't brake you can get even faster to places (although that wouldn't be very useful I guess).
According to google the universe is 93 billion ly wide. If you accelerate (and decelerate later) at constant 1g, this trip takes 49 years for the astronaut!