Having lived only 17 years so far, to ask what I expect to live to see is to cast a long, wide net.
When looking far into the future, I find it a useful exercise to imagine oneself as a non-human scientist: an alien, a god, or some other creature with a modern understanding of mathematics and physics, but no inherent understanding of human culture or language, beyond what it can deduce from watching what happens at a high level. Essentially, it looks at the world "up to isomorphism": it is not relevant who does what, what it's called, whether it has five fingers or six; but rather how much of it there is, whether it survives, and where it goes.
From this perspective, a few things are apparent: We are depleting our planet's resources faster than they can be replenished. Most of the sun's energy is reflected back into space without being used. There are more of us every minute and we have barely the slightest hint of slowing growth, despite overcrowding and lack of resources. We are trapped in a delicate balance of environmental conditions that has been faltering ever since we began pulling hydrocarbons out of the crust and burning them in the atmosphere (by coincidence, perhaps?), and there seems to be a good chance it will collapse catastrophically in the next 100 years if we don't run out of hydrocarbons first. We have thrown countless small, special-purpose objects into space, and some have transmitted very valuable information back to us. For a short period (while it seemed we would destroy our planet with deliberate nuclear explosions and immediate evacuation might be necessary) we played at shooting living men into the sky, but they have only gone as far as our planet's moon, still within Earth orbit, and sure enough wound up right back in our atmosphere. I should note here that I do not mean any disrespect to the achievements of the Apollo program. In fact, I believe they are among humankind's greatest—so far!
If civilization is to continue expanding, however, as well it shall if it does not collapse, it must escape the tiny gravity well it is trapped in. It is quite unclear to me how this will happen: whether humans will look anything like the humans of today, whether we will escape to sun-orbiting space stations or planetary colonies, but if we expand, we must expand beyond Earth. Even if environmentalists succeed in building a sustainable terrestrial culture around local farming and solar energy, it will only remain sustainable if we limit reproduction, which I expect most of modern society to find unconscionable on some level.
It has always been not only the human way, but the way of all living things, to multiply and colonize new frontiers. What is uniquely human is our potential ability to colonize all frontiers: to adapt our intelligence to new environments, or to adapt environments to suit ourselves. Although the chaos of a planetary atmosphere filled with organic diversity is a beautiful and effective cradle of life, it is no place for the new human-machine civilization. By some means—genetic engineering, medical technology, brain scanning, or something even more fantastical—I expect that humans will gradually shorten the food chain, adapting to use more directly the energy of stars. Perhaps we will genetically modify humans to photosynthesize directly, or implant devices that can provide all the energy for the necessary chemical reactions electrically, or scan our intelligences into solar-powered computing devices. Again, the details are very hard to predict, but I believe there will be some way forward.
I'm getting ahead of myself, so let's come back to the present. There is budding new interest in the development of space technology, in large part undertaken as private ventures, unlike in the past. Many view such operations as absurd luxury vacations for the super-rich, or at best as unlikely schemes to harvest fuel on the Moon and ship it back to Earth via a rail gun. I believe this research is tremendously important, because whatever short-term excuses may be found to fund it, in the long run, it is absolutely critical to the future development of our civilization. I also don't mean to imply that we should give up on environmentalism and sustainability, and just start over with another planet: these principles will only become more important as we spread far and wide, beginning in each new place with even more limited resources and limited contact with home. Not to mention that if Earth can be saved, it would be a tremendous cultural treasure to preserve as long as possible.
I'm not as optimistic about interstellar travel as some (I certainly don't expect it to become practical in this century), but I'm also much more optimistic about the ability of human civilization to adapt and survive without the precise conditions that were necessary for its evolution. There are so many possible solutions for the survival of humans (or posthumans) in solar orbit or on "inhospitable" planets that I expect we will find some way to make it work long before generational or faster-than-light voyages to faraway star systems; in fact, I expect it in my lifetime. But someday, "escaping the gravity well" will mean not that of Earth, but that of our star, and then humankind's ship will at last have...gone out.