Another way of moving over large distances without actually travelling that distance utlizes wormholes, also referred to as Einstein Rosen bridge. Wormholes are tunnels connecting one point in spacetime to another. They can make locations that are actually billions of light years away much, much closer. They create holes in the fabric of spacetime and bend it so the two ends are next to each other. Wormholes sound very efficient and useful, but we may never attain the engineering ability to capture, enlarge, and stabilize wormholes. If wormholes exist, they exist on very small scales, in the quantum foam, where they constantly pop in and out of existence. Because these wormholes are so small, they are impossible to detect with current technology. But let's say we could detect them and capture them. Then we would need to apply a source of negative energy to open them up and make them big enough for human-scale objects to fit. But we currently know of no sources of negative energy. The physics behind wormholes, though it may be hard to admit, is simply not there.
Lastly, a small comment about hyperspace/hyperdrive. Getting a spaceship to travel a couple thousand times faster than the speed of light is very implausible. Teleportation, which on the other hand is much more plausible, is the method that I feel is most likely to be the one we use to travel large distances in short amounts of time in the future.
Sources:
http://www.scifiideas.com/writing-2/writing-advice/6-alternatives-to-faster-than-light-travel/
http://www.popsci.com/article/technology/will-wormhole-travel-ever-be-possible
https://galileospendulum.org/2015/01/26/why-wormholes-probably-dont-exist/
http://discovermagazine.com/2014/julyaug/20-the-ups-and-downs-of-teleportation
http://www.express.co.uk/news/science/602529/Human-teleportation-is-possible-and-your-great-GRANDCHILDREN-will-do-it-claims-scientist
https://www.washingtonpost.com/news/innovations/wp/2014/06/05/quantum-teleportation-is-just-the-latest-mind-blowing-innovation-made-possible-by-breakthroughs-in-subatomic-physics/
http://www.popsci.com/article/technology/will-wormhole-travel-ever-be-possible
https://galileospendulum.org/2015/01/26/why-wormholes-probably-dont-exist/
http://discovermagazine.com/2014/julyaug/20-the-ups-and-downs-of-teleportation
http://www.express.co.uk/news/science/602529/Human-teleportation-is-possible-and-your-great-GRANDCHILDREN-will-do-it-claims-scientist
https://www.washingtonpost.com/news/innovations/wp/2014/06/05/quantum-teleportation-is-just-the-latest-mind-blowing-innovation-made-possible-by-breakthroughs-in-subatomic-physics/
- Ata Numanbayraktaroglu