Well, if you think about it, it will be a "stars or bust" situation to humanity at
some point. Meaning that at some point, we need to either lie down and die, or leave for a new solar system.
Even before that, Earth will grow too hot - Mars and the moons of Jupiter would offer some refuge for some time, but eventually Sun will die and that'll be the story of humanity, unless interstellar travel is developed. The stage where Earth will be rendered uninhabitable (oceans vaporize into the atmosphere) will happen as "little" as 0.5-1 billion years in the future. At that point, Mars will have pretty pleasant climate as far as temperatures are considered, but the atmospheric pressure would still be a big problem. Moons of Jupiter and possibly Titan of Saturn will be the last sanctuaries for life in solar system. Of course, living in Jupiter's radiation belts would be... interesting, to say the least.

But the gist of it is that in 4-5 billion years, sun will eventually expand to red giant and then either slowly wither away or blow it's outer parts into space as a planetary nebula. AT that point at least we need to have those colonization ships at ready - although preferably sending a lot of them away during longer period of time would be smarter than making them go at the same time as one huge rag tag fleet. All that about putting all the
eggs in the same basket beer to the same bag and so forth, you know... Perhaps groups of five to ten ships would deal with the travel time better than single ships, as far as psychological, social and cultural effects of generations long space travel are considered.
As to how actually construct these hundreds or even thousands of ships for the humanity (even in lessened numbers from present), I would recommend carving tunnels, living quarters, storage space, fuel tanks, facilities to repair or manufacture any part needed in the ship's tech from raw materials, hangar for maintenance ships,
fighter planes (who's gonna get into outer space without a fighter complement?!

) and of course transport ships to move from ship to ship in each group, a BIG greenhouse to make food with, in asteroids of sufficient integrity... and then slap some kind of engines on it's surface, as well as radiators for the reactors needed to offer energy for high pressure natrium lamps needed for the greenhouse(s).
Basically, a Rama of sorts, but possibly without the centrifugal gravity.
Meh, I'm rambling again. But it's strangely comforting that at some day, humanity will either have to at least try and get away from solar system or lie down and die. And knowing humanity, they will at least try, which will IMHO be the greatest adventure of humanity whether or not it fails or succeeds and to what extent. Who know's, perhaps a ship will reach a new habitable world... and they're gonna call it "Kobol"...

It's just a shame we won't be there to see it.
