Why would we journey to another planet?
We would go so that we would not die if/when the Earth becomes uninhabitable. If we don't leave the solar system that will assuredly happen.
Would we do the Star Trek thing? No, the current physics does not support that. The physics does not really support any travel to new planets outside our home system. There are theoretical transportation types, but for the most part they are impractical.
How would we move out of our system? I think that we have a couple technologies that we need to develop, to perfect before it is really possible.
1. A system of travel needs to be perfected. It does not need to be fast, but if it were faster that would be better, but either way, the transport would have to be able to de-accelerate and that is the trouble now. The craft must be accelerated beyond the escape velocity of the solar system and directed beyond to e next system to be colonized. It must be able to navigate the space in between stars and pass unsuitable stars and planets. This means a great deal of velocity and this means a great deal of braking as well. What is the use of getting off planet only to make a crater on the other side.
2. The quantum computer. The quantum computer runs with atoms acting as the bits of information. A very small bit of matter would be able to act as a computer and storage device capable of storing every chunk of data created by humanity. It would be able to store that and the code for every kind of molecule and DNA pattern on the face of the Earth. It can do all that and build and run very complex machines that could terraform any sizable chunk of mass that it would encounter. And that leads us to the third component
3. A cornucopia machine. This theorized machine uses a quantum computer to build whatever we need. It is not as fast as Star Trek's replicator, but it is more versatile. When the space craft lands it will start replicating nano machines, the nano machines will start mining the landscape and start building power generators, solar likely. The power will then be used to creat more nano machines. The nano machines will hit a critical mass and start building micro-organisms that can exist on the mass. If the mass is too small, it will start building enclosures so the organisms can start to change the environment. The machine will modify the environment, then it will start building artificial wombs and start creating artificial humans for the wombs, robots to see to the human's needs and education, it will start to build food for the humans to eat.
The problem with space travel is that it is incredibly expensive. To put a person in orbit one has to put everything that the human needs to survive for the duration of the trip. For a week this is not much, a seat, food and water, a supply of a supply of breathable air, shielding from the dangers of space and a really big diaper. For several months to years, the human needs all that and space to keep fit mentally and physically and lots of power to maintain it all. All this requires mass. The energy needed to move mass is the same just proportional to that mass.
To move a small rock out of the solar system and slow it down when it gets where it is going is beyond our current understanding, but to move a million times that mass to the next star, is a million times beyond our understanding.
So for humans to leave the solar system, we will have to breed machines that will build us, once a livable place has been built on other planets in other star systems.
One really neat side effect of this method of spreading humanity through the galaxy is that on each New Earth, humanity would evolve separately from the others and if they eventually did reestablish contact with each other after a few thousand years, it would be like Star Trek, hundreds of near humans that can breed with each other; the same form evolving on hundreds of different worlds
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