This is one of the most fascinating questions I have seen in some time on this site because there are so many dimensions to the question.
(I am being rather America-centric here but considering the needs of your astronauts - a highly developed space agency, civic infrastructure such as roads, instruction manuals that every astronaut regardless of their origin can read, ease of access to a wealth of resources, safety in terms of wild animals, clean air and water - The USA and to a lesser extent Canada would be a very good choice. Russia would be excellent as well though, and I think most everything I mention here would still apply.)
Specialization NASA astronauts, and I assume current spacefarers from the rest of the world, are chosen for a given mission based upon the experiments and repairs that we intend to execute during that mission. So, you need to consider what these six are good at. You must have at least one pilot and one engineer, probably more of each. Chances are you have a biologist/geneticist who may be an MD. This would be very good for your scenario. You probably have a chemist. Pretty much everyone will be a decent level computer programmer. You may also have a botanist if you are lucky. A lot of prolonged space-based experiments involve growing various plants in zero-G. Whatever you have, they are six individuals of high intelligence in extremely good condition and with very advanced, but fairly specialized knowledge. The practical end result of this is that there will be significant gaps in their knowledge. Let's discuss later where they will be important.
Reproduction This has been covered well. An immediate birthing program would need to be put in place, but I think chances are, the astro-men would have a rather limited role here. Their sperm would remain viable for many years to come, whereas they may be in a hurry to find viable donors of the greatest genetic diversity possible, both male and female. I don't see a whole lot of issue here worth noting. The biggest source of conflict is likely to be that we have a viable Mongolian woman's egg, and just the right South African father's sperm, but the blood types don't match any of the mothers so you have to trek to find the correct anti-rejection drugs. Also, the first several generations of girls will need to start getting pregnant at or near the age of 13. Pregnancy and sex may become vastly separated concepts.
Disease Here's the good news. The plague killed off almost all sources of infectious disease your hero's are going to run across. Both viruses and bacteria tend to be highly specialized and those that will harm humans tend to live only in humans. I say 'tend' on purpose. There are still many that will cross species boundaries, and mammals are going to be your biggest danger with birds close behind. Parasites can be a problem, with malaria being a bigger issue than the common cold to your heroes. Forget the flu and pneumonia, lyme disease and rabies is your killer.
Education This one is tricky. What good will a sociology or pharmaceutical (linguistics?) degree be to generation 0-10 humans? The knowledge would be preserved through books and video lectures, etc., but at what point would you decide that this is knowledge worth having? At such a point, who would be around to teach, or can people have enough general knowledge that they can self-teach? Would such pursuits be planned out by Gen-0 so that Gen-10 has a job assigned to them at birth/first-appearance-of-aptitude? Are they simply lost and rediscovered 1000 years hence?
Some things may seem unimportant at first, and in fact will rely upon skills our astronauts do not have, such as farming. There will be tons of food laying around, and a lot of it would survive for decades, though your choices get fewer and less interesting as time goes on. And six non-experts working full time do not have the ability to raise the biodiversity necessary to make a good lasagna from seed. They would still have to. There are seed banks out there, but they need working refrigeration to last more than a few years.
More on Food Since mammals and birds will be the biggest bio-threat to your heroes, fish would be the best initial source of sustainable protein. It would be a good idea to husband/domesticate a few reptile species as well, though choosing them would have to be done carefully. Iguanas come to mind, and constrictor type snakes that don't get too big would be good. Even boas and pythons that are slaughtered before they get to be around one meter long would be ideal as long as they procreate before that. Wild versions of big-agro plants would be abundant for generations, though the American food-belt would face drought conditions for generations without anyone to maintain irrigation systems.
Security Let's face it. All animals are wild animals now. At the point where people realize there is no bunkering down and surviving this, they will let their pets loose. After an entire year, they are pretty much feral. But after only a year, some will return to domestication with little effort. Dogs and cats may become as important as they were to the cavemen. I ascribe all malicious intent to humans, and we can consider that gone at this point. A solid, well-maintained chain-link fence three meters above ground and one meter below completely surrounding your new home should be fine. The materials are readily available and it wouldn't take more than a week or two to secure a sizable area for several generations.
Keeping the power up That oil/gas/petroleum we were worried wouldn't last more than a few decades? We now have enough for several hundred years. The electrical grid will be hugely unreliable though. A storm could take down a few lines, and if the option is between finding the central office, turning off the proper switches, getting a cherry-picker to the right location, repairing the line yourself and switching the power back on with all fingers crossed, or simply moving to an unaffected area, your heroes are probably moving (Or just living without a power grid). I think localized power sources will dominate. Wind and solar. After a few generations are out the shoot, I wouldn't be surprised to see someone develop a hydro-dynamo in the small-scale. Gas would probably only be used to power cars and for heating and cooking. Viable seed and reproductive cells would have to be moved to where the heroes could guarantee continuance of power. Maintaining this system would be a constant concern, if not a full time job.
I think the interesting part would come in 20 generations into it when two-thirds of the population are forced to found new population centers, or 30 generations down the line when the all-knowing computer program which the progenitors set up says it is time for governance and law enforcement. Or 50 generations in when someone approaches the machine to receive their job and 'Archeologist' comes up, and no one knows what it is.