I'd start with an evolutionary account. AO Wilson, the leading biologist on social insects, developed and extended the idea of eusociality, to account using multi-level selection for the emergence of hive organisms. And his account also applies to some extent to prairie dogs, wolf packs, naked mole rats, and human tribes through the majority of our evolution (whether we have group-regulation of childbearing now is disputable, so it could be argued we are no longer eusocial). Crucially language capacities developed during a eusocial era, and we can look to Dunbar's Number for evidence, and to understand how it's driven our late-developing neocortex, which we essentially 'program' to inhibit impulses societally.
Then, how did morality evolve. The case of why humans, almost uniquely among species have sex in private, has insights. I gave my views here How do ethicists tackle the question "Is it immoral to have sex in public places?" Is it possible to use rational and empirical ideas to answer? The takeaway for me, is evolved shame, disgust & pride, have been progressively hijacked to alter social behaviour unconsciously, and increasingly, consciously.
I think humour is an interesting example case. The closest analogue we have to human laughter is the chimpanzee pant hoot. We see tickling in chimpanzees, rats and humans. We see play in many animals, especially when young. And we have the self-domestication theory of humans having extended childhoods and neuroplasticity, and keeping some features of juvenileness like playfulness, as supporting our intelligence. In a social context, I see the role of humour as easing social tensions (like sex in private also does). Humour allows competing concerns, and boundary enforcements, to be done in playful ways that don't threaten status. We have taken these evolved behaviours for infant play, already hijacked for bonding through (infectious, like yawning) laughter, and hijacked it further to allow space to play with serious subjects that isn't threatening.
Next, sociology. Durkheim has a great account of religious practice, that it's about binding communities together through holding things sacred, beyond question, and enacting, celebrating, and enforcing that. This perspective allows us to see habeus corpus, and free speech, as defining of communities as having an altar, or an ark of the covenant. Challenge habeus corpus, and you challenge the basis of coherence of the community bound by it (see Hong Kong as a live test case). Durkheim links social decoherence from loss of shared binding values, to personal anomie.
My favourite anthropologist is David Graeber, and his book Debt: The First 5,000 Years gives a compelling account of the emergence of trade networks and currency, from mutual obligations and trust-building.
Then I think we need Nietzsche, and the post-modern challenge to society-wide metanarratives. The best famous example being 'god is dead'. For most of human history, conformity to religious verities, like say the literal truth of the transubstantiation, was enforced by threat of death. We can see why, using Durkheim's framing - challenge cultural truths, & you challenge cultural cohesion. I'd say strongly correlated to the printing press, that approach of threats largely doesn't work any more - you can't just force a society to take up views any more, to, essentially, bind it to one book (which historically tended to happen when the monarch converted). But this disputing about how to live, what values to hold, challenges social coherence. We have a certain set of 'enlightenment values' that a good chunk of the world hold, and celebrate, and try to enforce, and that underpins the global community. The challenge to that social hegemony of values, is a challenge to that communities binding power, as the global communities values are a challenge to what binds China who's government largely does not share them. When we make a cultures metanarrative intelligible, reasoned, it can reach beyond borders - like the scientific community. We need challenge, rethinking, reform, improvement, of sets of metanarratives, but we also need their binding power, so it's crucial to find the right pace of change, or a bloody revolution gets followed by collapse & chaos.
It's interesting to look at how the picture of 'human nature' affects a person's view of society. Rousseau pictured a 'noble savage', saw civilisation as corrupting, and saw the social contract as between mature intelligent people. Hobbes saw the state of nature as 'the war of all against all', and the pact of society as forged out of fire and blood, and being to submit to the lineage of the strongest, the monarch, to avoid more fire and blood - he lived through the English Civil War, and the beginning of witch panics, so we can have some sympathy. Jonathan Haidt has some interesting insights with his moral foundations theory, on how feeling threatened through our key development age (under 25, neocortex still developing) shapes our political outlook by suppressing tolerance of ambiguities. Also herder 'honour cultures', vs the ethics of rice & wheat growers who must plant and harvest together. I see the Left as arising out of eras of prosperity and security, and motivated by what they move towards. The Right as arising out of eras or regions with turmoil, collapse, and motivated by what they move away from.
Permaculture, is a modern school of thought, properly called a design methodology, rather than a philosophy. It aims towards generating 'permanently sustainable culture', so is premised on recognising a lot of what we do can't last, like if everyone on Earth lived like in x country it would take y times the Earth's resources. So, it becomes about conscious culture design. It isn't a set of answers to how to design cultures, but an approach towards how to do it. I take inspiration from The Arts And Crafts Movement, and Japanese tea culture & wabi sabi concept, to see tgat consciously chosen aesthetics and a kind of design manifesto, can do more to shape how we actually live, than a lot of the disputing about what is true. Rudolf Steiner wanted to abolish squares, rectangles & regular circles & triangles in architecture, to challenge the cubes of cubes in grids aesthetic of the then industrialising world. Gaudi's approach to architecture used catenary arches, and much else inspired by nature, which we can see feeding into the idea of 'biophilia', being brought to bear now on urban design, like closing many streets in Barcelona's street-grid to fill them with trees.
Some tools for a toolbox there, on thinking about shaping society, and what culture is, and why it matters.