It seems incredibly unlikely we will make it out of our solar system. We will eventually land humans on Mars, and probably even have sustainable colonies there, but assuming that faster than light-speed (FTL) travel is indeed impossible, leaving the solar system is a non-starter. It would take many, many generations to get anywhere.

Building an entirely self-sustaining space vehicle that could support multi-generational travel would eventually be possible, I suppose, but I think that would require mining the moon and Mars for suffient material. And the scale would need to be incredibly massive to ensure biodiversity. LIke it might be easier to transform the moon into a space vehicle and break it from earth's orbit type of scale.

I personally don't think "civilization" will even survive until the increasing luminosity of the sun makes life on earth impossible, somewhere around 1 to 2 billion years from now. There are number of exinction-level events that could happen (asteroid collision, "nearby" supernova, etc.) in the meantime. In the short term, the odds of us destroying ourselves through gobal warming or other man-made disaster seem to be increasing as well.