There are a few things about space travel and science that it bugs the heck out of me that we don't know.
Astrobiology is the biggest one. We do know that one cannot live without gravity forever without at least some degree of medical help. What do we need to do for a person to live in freefall long-term?
If some amount of gravity is necessary, is rotational artificial gravity acceptable or is there something else at play? We've put people in centrifuges on earth for long periods of time, but we've never spun somebody up to 1g for long periods of time. On earth, there's a limit to how many RPMs you can stand before you get sick, but we don't know if that also applies outside of the downward pull. Is exactly 1g a requirement or is there range of acceptability? Is there one range for personal survival and a tighter range necessary for being able to bear a normal child to term?
To me, this means that we need to breed many generations of model animals in space under various gravitational and centerfuge gravity situations and match them to a control group on Earth.
What is the nature of Space Adaption Syndrome? Given that for maintenance tasks, it's largely simpler to work in zero gravity, if it turns out that it's better for people to return to gravity at the end of the day, can a space-beamjack switch between environments? Is this a level of susceptibility that can be tested for? Is it better for workers to spend six month shifts in zero gravity, only returning to either earth or a space colony at the end of a shift?
There are also fairly untested and un-attempted ideas.
For example, there is the Space Activity Suit design that replaces all of the complexity of the space suit with a spandex bodysuit that provides pressure to the body. This might make it much easier to work in space.
Or progressively more sophisticated environmental system. Presently, we just installed in the ISS a set of equipment that will recycle urine into drinking water, which is trickier than you might think. Otherwise, CO2 is dumped overboard with molecular sieves, water is split into O2 and H2 where the H2 is dumped. There are some thoughts of combining the H2 with CO2 to make hydrocarbons and more oxygen. We can spend all the time we want on Earth talking about it, but the real test is to actually fly it.
Nor have we tried to fabricate a partially or fully biological regenerative life support system and fly it in space.
We have not tried to even do a simulated long-duration mission. Such a mission would involve spending at least a year without resupply. Both Mir and the ISS have regular flights up and fairly frequent spare parts and food being sent up.
We have not done sophisticated troubleshooting and repair in space. The basics of the idea have been investigated, so we've tried soldering and welding in space, but NASA always drags parts down for troubleshooting. The farther you get from Earth, the more the ability to jury rig and re-manufacture becomes a survival task once you've swapped out your last spare.
We have not tried to remediate the radiation environment of space. NASA's astronauts to the moon took a respectable dose of radiation. Present missions seek to reduce the risks by controlling the amount of time in space and living in low earth orbit with a more reasonable dose of radiation. This won't work for longer missions. There's cosmic ray activity and such to consider.
We have not done deep engineering work on any of the nuclear or plasma propulsion drives that would allow us to reach other planets in a reasonable amount of time, instead of taking low-energy transfer orbits. The same goes for space-based nuclear reactors built to a reasonable production specification that might allow us to have power through the Lunar night. We may fly VASMIR in 2011.
There are proposed missions that we really ought to do, for a number of reasons, but haven't. One of my pet missions was the "Thousand AU" mission. See, beyond a certain distance, the only way we know how far a given object is from us is because of Hubble's Constant and the measured redshift. Is it right? We don't necessarily know because there's a lot of things that could screw up our map of the universe. Given that a LOT of our knowledge of physics is based around what we've observed, that's probably not always a good thing. ESA's Hipparcos mission did a decent job of improving this measure from orbit. TAU was going to use an ion drive to get very far from Earth, enough that we could measure most of the galaxy via paralax instead of just redshift.
If you check through NASA's technical reports, there's tons of stuff that's been proposed, questions that have been asked, etc. and we've never tested the ideas or answered the questions. It's kind of like the air guitar version of space science.