It won't be easy. Each settlement in the solar system will have to be built with total reliance on technology. How long would it be before the solar system is as crowded as earth right now?
I won't worry about what is next after the solar system, there is simply too much to be done just with that, and it won't happen quickly. The solar system should be developed, it belongs to us as much as earth. And, I don't care if the Chinese colonize the moon and Mars, there is enough for all for a long time to come.
If we can figure out how to get to other stars relatively easily, through new discoveries in physics, then we wont need to waste our time developing inhospitable planets, but can search around for more hospitable ones...Earthlike ones. Much would depend on just how rare a planet like Earth is.
But I suppose we need to crawl before we can walk, and walk before we can run. Just overcoming the problems of gravity would be enough to get us started.
Tuor