With the recent discovery of a possibly Earth-like planet some 20+ light years from Earth this is a subject that will be given thought by many people (especially those who would like to build a civilization from scratch). Given many more thousands of years of technological growth at the current rate it may one day become possible for humans to colonize other planets. However, considering the numerous obstacles involved in undertaking such a project, it may not be practical to even make the attempt.
The first and foremost problem is where to colonize: in the solar system or outside the solar system.
If there is definitely a habitable planet outside our solar system there is a major roadblock to colonizing it. Even with thousands of years of technological advancement it is doubtful that human beings will ever be able to travel (in a reasonable amount of time) outside the solar system unless some discovery in physics invalidates our present understanding of the Theory of Relativity. The speed of light is absolute and the energy requirements to achieve something even approaching such a velocity would be astronomical and totally impractical. The only other two options would be to either have some kind of suspended animation ship or some kind of multi-generational ship. A suspended animation ship would be the best choice from an energy conservation point of view but by being completely automated there would be no guarantee that the ship would reach it’s final destination. A multi-generational ship would have a better chance of reaching another solar system but would the descendants of the original colonists even care about living on a planet. Unfortunately, a multi-generational ship would need a tremendous amount of initial resources making such a ship somewhat impractical.
Choosing a destination within the solar system would be much more practical but there is one major problem, there are no planets within the solar system habitable to humans. Any planet or moon in our solar system would need a lot of renovation either by building an extensive network of habitable facilities or by terraforming the destination. Either of these options would be an inexcusable drain on the resources of the Earth and would be completely unacceptable to any future conservationists. Unless it could be shown that the Earth would definitely become uninhabitable I doubt anyone would stand for the Earth being stripped of resources for a new colony.
Another major obstacle involved with colonizing another planet isn’t so much technological but societal. Historically there have always been divisions in society and divisions continue to exist even to this day. There are many people nowadays who would probably like to go off and colonize another world to escape the world we live in and there will probably be people in the future who will feel the same way but as long as divisions in society exist no project of such magnitude will ever have a chance to succeed. A colonization effort is going to take a tremendous amount of resources and it is doubtful that the people of the Earth are going to want to sacrifice those resources to be rid of people they do not agree with. Colonists 400 years ago had access to what little resources the ocean and weather provided but colonists crossing the vacuum of space will not have access to any resources so they will have take everything with them.
If society evolves enough to eliminate the divisions that obviously exist there is a possibility that a colonization project would succeed but would the desire to do so still exist. If societal divisions become nonexistent then it must be assumed that the problems in the world have been solved. If there are no problems in the world then the people probably will not have any reason to leave except possibly curiosity. Curiosity is a strong motivator but is it enough to overcome the depletion of Earth’s resources?
Will humans try to colonize another planet? It is so far in the future that there is no way to know.