Is it possible to effectively travel faster than the speed of light? In his last book, Brief Answers to the Big Questions, Stephen Hawking states that:

If one couldn't go faster than light the round trip from us (...) to the centre of the galaxy would take about 50,000 years. If the spaceship went very near the speed of light it might seem to the people on board that the trip to the galactic centre had taken only a few years. But that wouldn't be much consolation if everyone you had known had died and been forgotten thousands of years ago when you got back.

This radically changes my understanding of special theory of relativity. I always thought that you can never reach areas of the universe that are too far away (say, thousands of light years) in your lifetime. But what Hawking wrote seems to imply that you could effectively surpass the speed of light, because time slows down for you (even though it would seem that the trip took you a longer time for an outside observer). Is Hawking right?
 A: Yes. A photon doesn't age. You would age very little compared to the people left behind on Earth you if you were traveling with almost the velocity of light. You also effectively cut down the distance due to space contraction. 
But problems with that far outweigh the perks.
For example, even the vacuum of space has some density of particles and fields, albeit a very small one, but at that speed, the resulting "drag" could still be quite brutal. 
A: Yes. Einstein and Hawking are both right, but you don't travel faster than light.
On your high speed journey, time for you passes much more slowly than it does for the observers you leave behind
You can reach your destination as soon as you like (almost), but you leave the rest of us behind for ever (almost)..  If you could reach the speed of light, time would stop for you and you would be there in an instant.
That raises lots of follow on questions.  The short answers are:
While you are traveling it looks to you as if our time is passing more slowly than yours.
When you turn round to come back, as your speed slows then reverses, our time seems to you to race into our (almost) infinite future.
When you get back you probably won't recognise the place.
It's all weirdly fascinating, and proved bit by bit by small scale experiments.
A: You can never surpass the speed of light in a vacuum. Light has no rest mass so nothing with any mass can travel faster. Light does slow down when it enters a medium (for example water, which causes refraction) so light in the surrounding air would then be travelling faster than the light in the water but its only these types of cases which "break the rule". 
To accelerate an object with mass (a spaceship) to even near light speed would take more energy than we can imagine producing. There was an idea once to drop atomic bombs out the back of a spaceship to produce a lot of acceleration but I don't think anyone ever really tried it (maybe in Area 51 - just joking). 
The thing to remember is that if we keep trying it might someday be possible to achieve real space-travel but no matter how fast you go light inside your spaceship will still be travelling at the speed of light. If you pointed a torch away from you at that speed you would never see the light from it. If you went faster than light you could arrive somewhere before you left. 
A: Also in this case of comparing times (or aging) after return to earth,
the speed can not be the (only) reason for the different aging.
If an austronaut is passing by the earth with constant speed vector, it is not possible to know the absolute speed of earth and traveller, only the relative speed is known. F.e., both could move in opposite directions with half speed, or the earth could leave the astronaut. 
The only difference, i.e. symmetry breaking, is the acceleration resp. gravity field (180 degree turn around a big mass) that the astronaut is experiencing.
If the earth experienced the same trajecory resp. acceleration/gravity field as the astronaut (f.e. mirrored travel path), there would be not any difference in aging - just for symmetry reason. 
