Colonizing the galaxy is a political problem, not a question of engineering possibility. A sufficiently zealous world government could jump-start asteroid mining with an Orion Heavy Lifter, construct mirror arrays near the sun, and start lobbing around interstellar VNMs, all with relatively simple refinement and application of existing technologies. Problem is, the best way of putting a complete industrial base into orbit runs afoul of certain atmospheric nuclear test ban treaties.

Without a reactionless drive there's no point sending a colony ship faster than about 60% of the speed of light. Gotta save some remass to decelerate.