At that speed and assuming a lifespan of 60 years. The answer depends on the motions you include. The speed of the solar system around the Galactic center is about 230 kilometers per second. If you only include that, then you're traveling 726 billion kilometers a year, or 479 billion kilometers in total.
If you want to “get out more” in stellar terms, consider inventing an antimatter engine that can take you up to 99.9 percent of the speed of light (and slow down again). If you wanted to travel with a constant acceleration of 1G and then a deceleration to the very edge of the observable Universe. That amount is so small that you'll never notice it, but if you're traveling at close to the speed of light, the differences build up pretty quickly. Space is crushed in the direction in which you travel, so, for you, you didn't travel as far as the stationary observers saw you traveling.
They may only travel 71 light-years from space, but eons would have passed outside the universe. However, the really strange consequence is that, from their perspective, thanks to relativity, flight times are reduced. If someone on Earth lived 71 years, someone else on a ship traveling at a speed close to the speed of light would only experience a small fraction of that time. If you count this as your own journey, you will have covered some 62 billion kilometers in 66 Earth years.
If you accelerate at that rate for years, you can travel billions of light years over a human lifetime. Individual human beings don't live billions of years, so they couldn't travel billions of light years at the speed of light. However, we must also add the distance that the Earth has traveled around the Sun and the distance that his house in London has traveled around the Earth (at a speed of about 0.465 kilometers per second). With all due respect, Fraser, the average human lifespan (global average) is only 71 years, so the longest distance that could be traveled while traveling at the speed of light during an entire human life would be only 71 light years, not billions of light years.
It has nothing to do with “time travel”, but only with the reference point from which the measurements are made. So, when you approach at the speed of light and look out the window, you see that your destination is much closer.