How will humans survive the radiation?
How will humans remain mentally healthy?
I don't think we can feasibly become space faring until those challenges, and probably some others, are addressed.
Edit: Oh, I just had an idea. How do people with Asperger's deal with isolation, boredom, continuous contact with a handful of other people? Maybe there's a subgroup of people who are mentally perfectly adapted for space travel.
Q: How will humanity survive the radiation?
A: Build igloos or inhabit caves or just plain dig into the ground. Problem solved.
Q: How will humans remain mentally healthy?
A: The Internet and social interaction with other colonists. Also, there is a lot of work to be done growing plants, managing systems, exploring robots, documenting and cataloging all that we find.
People, today, already live in extreme isolation for extended periods of time by choice.
Those aren't even the hard problems to solve.
I'm more worried about the time in transit.
For sanity, my assumption is that humans will need the ability to easily increase / decrease metabolism. So that 6 month, year, century trip would seem like a few days.
For radiation, I think space faring humans will need to be radiation hardened somehow.