Radiation is a huge problem. Both en route and on Mars. A lot of these problems beg the question. Is Mars really the best target? Sure it's close but there are moons around Saturn that are interesting as well. Titan for example. Sure it's cold but it has a lot of atmosphere. Presumably landing would be easier. If we could learn to live underwater we could possibly visit Ganymede or Europa.
Human psychology is another, perhaps even bigger problem, also independent of destination. We've had various experiments with prolonged isolation, but none where the subjects know that they're never coming back to Earth. I'd say psychology has a >50% chance of wrecking the first manned missions to other planets/moons, unless we send huge ships with hundreds of people.
In the Red Mars trilogy by Kim Stanley Robinson, the author has come to the very astute observation that
a) no perfectly rational, mentally healthy human would volunteer for such a mission
b) there will necessarily be extensive psychological screening to ensure we only send rational people
thus c) the people who will be selected are likely to be non-rational actors very good at concealing their non-rationality from psychologists. Also known as people with psychopathic tendencies.