Who knows. There are a million variables to consider. That said, I believe the most likely scenario is as follows:

Around the year 2060 a strong artificial intelligence will be created. If we are wise and have conducted adequate testing, the artificial intelligence will turn out to be a benevolent one and will begin pursuit of the utilitarian ideal of maximizing good feelings for the maximum amount of people for the maximum amount of time. The machine will realize that to achieve this ideal, it will require a lot of energy and space. What is the greatest source of energy that is potentially within reach? The Sun. And so the machine will set out on a quest to tame our star. It will construct all manner of robots and rocket ships, utilizing a great amount of resources in the process, that will exist for the purpose of establishing an industrial base on Mercury. The machines planted there will harvest Mercury's resources and convert them into solar-harvesting satellites to be put in orbit around the sun. This may be the beginning of a dyson sphere. With a large and steady flow of energy, the AI will create ever more machines to begin the terraforming of Mars and perhaps other planets as well. Hopefully the AI will figure out warp drive around this time so it can begin expansion to more hospitable planets far outside our solar system.

However, humans may not have much of a part to play in space. The thing is, any mechanism for fast-travel we uncover is likely to be extremely volatile. If you've read about the possibility warp drive, you also know both that force fields may be possible and that they would be required for high-speed travel in order to defend against space debris. What would happen if a spaceship with this kind of technology were to slam into an inhabited planet? It could cause an apocalypse. Observing this risk, the machine would either prevent humans from flying spaceships or it would install extensive safety mechanisms in any spaceship humans are allowed to control. It doesn't matter if most people are good, all it would take is for one person among trillions to go insane to initiate destruction like the world has never seen before.

So let's see... If all goes well, in fifty years we create strong benevolent AI. It may take anywhere between 1-10 years for it to build up an industrial base and infrastructure to support its activities on Earth. After this, it could take God knows how long to establish industrial activities on Mercury, send satellites into orbit around the Sun, AND terraform Mars. In my scenario, I'd say at least 60 years for the ball to even start rolling. It could take another century for humans to be able to inhabit other planets. I think 80-150 years in the best case scenario is a good estimate.

I suppose in the next 30 years or so we could have some sort of moon base, but I wouldn't really consider this serious expansion, myself.