Forgetting the part that we would need literally an infinite amount of energy to get to the speed of light in the first place, time would stand still for someone travelling at the speed of light. So from the traveller's point of view, they'll arrive at their destination at the very moment they hit the speed of light.
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http://www.emc2-explained.info/Time-Dilation/
But let's say the traveller reaches a speed very close to the speed of light, and travels a 1000 years from our point of view. Then in their frame of reference they arrive much sooner, perhaps after their calendars, clocks and bodies have progressed 50 years.
This is also one of the (currently hypothetical) problems of interstellar travel: even if we manage to reach speeds close to the speed of light, and manage to travel to another star and maybe even back....the travellers may have survived the trip....but here on Earth a much, much larger time has passed. When the travellers return, their family and colleagues have already died millenia ago. Perhaps even humanity doesn't exist anymore. A strange thing to contemplate. Someone should make a movie out of that :-)
I guess the best way to appear to be travelling close to the speed of light is by being able to exploit 'wormholes', time-space curves, etc.