Considering the high cost of propelling mass to another star system, there would be nothing extra being carried. Period. The ship would carry the minimum population needed to guarantee the success of the mission. And they would have the minimum facilities and supplies needed to guarantee the success of the mission. While there would have to be considerable margin over what is actually needed to guarantee success in unknown circumstances with no chance of additional support, apart maybe from moral support, from home, those people and resources would be a reserve, not something useless or available for luxury.
Everyone and everything on the ship would have a proper place and a proper purpose, a part to play in the mission plan. While there would be redundancy, possibly a high degree of it, it would be distributed equally, there would be no redundant people or supplies. This would guarantee a smoother transition, if portion of resources is lost and when the ship arrives at its destination and expansion of resources and population becomes possible and necessary for the mission. And yes people would be considered resources, not because people are not valued, but because even supplies such as air and water would be as irreplaceable as we'd like to think we think people to be. Waste and carelessness would be anathema to these people.
Every resource, human or otherwise, would be utilized in the most efficient available way. This might not be the one most efficient for the individual resource, rather effort would be made to see the big picture. The effects on other resources would considered, as well as the resources used on planning.
An entire population with a shared purpose that is provably true, precisely defined, and concretely present every moment for generations would be quite different from anything we are familiar with. Elite military units maybe get closest, but they have life before, outside, and after the mission. The shipfolk would have no outside to go, only the first generation would have life before, and only the last would have life after. Some religious fanatics may have that kind of pervasiveness, but they have the issues of faith to deal with. The shipfolk would know what their purpose is with precision and certainty that even the most dogmatic fanatic would have to fake.
The shipfolk would have no need to convince either themselves or each other of what they are doing and why, everybody present could just read the large scale detail from the mission plan whenever they want. And while there would be uncertainty and even disputes about the details, resolving such issues would itself be part of the mission plan with resources allocated and procedures established to prevent any serious risk to the mission plan.
Unless this failed, in which case there would be a real chance of everyone dying, the shipfolk might end up having no real concept of unresolved personal conflict and no concept of personal grudges, or decisions motivated by personal reasons. They'd still be human and feel all the personal stuff and act on it, but it would have no space to grow into anything significant. It would be just something to compensate for while doing business in a quiet efficient manner.
Pressure to be efficient and to have active redundancy would presumably make for relatively relaxed pace of work. Fast and efficient, but never rushed, apart from drills maybe. Combined with everything having a proper, well defined, place, I'd assume people would end up being polite, respectful, even formal, but in a direct and relaxed manner. Formalities would be observed, but little time would be wasted on them and a failure to observe formalities would be noted, pointed out, and forgotten. Protocols exist to make work smoother, they do not have any value of their own.
Still there would be considerable social pressure to act proper and given the "from birth" nature of the shipfolk society, it would be very deeply ingrained on people. Any lapses would be rare signs of stress or slightly more common attempts at humor and would be understood as such. Fortunately, the almost compulsive pragmatism of a society that actually has a clear specific purpose for everything would guarantee that acting proper would not be difficult enough to cause stress and distract people from work.
With a relatively high degree of planned redundancy and presumably high degree of automation, the routine work would only be a relatively small part of the lives. Much time would be spent in training since it would be much easier to have high degree of redundancy if everyone is competent in several different duties. People would also change duties relatively often, to keep that competence fresh and avoid boredom.
Drills would be frequent. The shipfolk society does not have much space for competition, so drill performance would the major outlet for competitive pressures and personal initiative and ambition. While periods for relaxation and leisure would be scheduled, drills and training might also be the dominant form of entertainment. As such considerable effort would be spent on making the drills varied and interesting.
The people would also have specific training for efficient relaxation, probably meditation of some sort, which would add by contamination a contemplative and spiritual side to the culture. The disciplined lifestyle would support this by making the shipfolk naturally relate well to monastic traditions.
If you add together the need to minimize the population on the ship and maximize the genetic diversity at destination, you end up with the ship carrying huge sperm banks containing genetic diversity equal to fairly large nation. Unless artificial wombs have been developed, the ship population should then be entirely female to allow for rapid population growth at the destination.
During the trip the population would be stable. Births would equal deaths. Families would presumably be built around mothers, daughters and sisters.