Your basic understanding of special relativity is correct. By travelling very fast, in fact, by travelling at velocities extremely close to the speed of light time passes slowly compared to "stay put" people and places.
Yes people with access to near-lightspeed spacecraft could travel away from the Earth and return to what is effectively its future.
Now, while the theory is simple, the practical application is mind-bogglingly difficult. For example, energetically a spacecraft of this sort would consume more power than the human species has used during its entire history to date. This means near-lightspeed spaceships won't be possible for a very long time. This also might be if ever. It is quite possible the human species will never build near-lightspeed spaceships. Travel to the stars may take place at lower, much more practical velocities.
The energetics question for near-lightspeed vehicles is only one of many problems that will need to be solved before this form of space travel is a practical possibility. There are a multitude of other problems all of which make it an effective near impossibility.
There are questions like like the economic cost. Who makes the choice of those sent into space in fast spaceships?
Plus it isn't often realized how close to lightspeed a spaceship will need to go to achieve large time differences during its flight, For example, a spaceship travelling at 99.5% of the speed of light will only have a time dilation of ten. This means if your couple to return to Earth two hundred years in their future, their travel time in the spaceship will be twenty years. This assumes the spaceship's average velocity will be 99.5% of lightspeed. Spending twenty years together in a spaceship might trying for even the most devoted of couples.
Since you are creating a fictional world of your own there is nothing to stop you, as its creator, to assume all the technical problems have been solved. Now you can have a future full of near-lightspeed spaceships and couples can effectively travel into own futures.
This a common assumption of many science fiction stories where special relativity and spaceships travelling at close to lightspeed enable their characters to go into the future. Most of that fiction isn't interested in exploring the details of this kind of travel, it usually assumes it can happen and leaves it at that. So can you. The basic principles are sound. While the science is good, the engineering is unbelievably difficult. Know when to leave well enough alone. Just let it happen and focus on the story about what your couple finds in the future.