You've created a world. The "people" there, human or not, don't live like we do. How do you go about writing their lives - their manners, their rules, etc. - without sounding either pedantic or overblown? It's not as easy as it looks, but I hope this checklist will help you to get a good start.
1. Identify social and cultural parameters
Some of you may already have done this, i.e. listed out the social rules by which the people of your world live. If you have not, it's worth doing. This single step in itself can be a long process, as there are a lot of areas in which cultural parameters operate. I'll list some of the areas, just to give you some inspiration. Greetings and manners, architecture, food culture (preparation and consumption), economic roles (work), taboo behaviors (verbal, physical), education, elites (economic, educational, etc), religion, folklore, etc., etc.
There's a lot out there, and if your world is comprehensive enough, it will be enough for your story to get lost in. To make sure that doesn't happen, make sure to...
2. Organize and prioritize your parameters
Of all the possible cultural parameters you might have come up with, not every one is going to be equally important to your story when you write it. Is there a particular cultural artifact, or set of assumptions, behaviors, or practices that are going to be central to your main conflict? Are any assumptions going to take a back seat, but still be important to themes of the work? It's a good idea to have a list, or at least a clear sense in your head as you start, what these parameters are going to be. Culture is so huge and complex that focus is really important.
3. Identify "problematic" parameters
The extreme version of a problematic social or cultural parameter is one that will be difficult for readers to accept. Perhaps they'd say something like, "I just can't believe that people would not want privacy here." Problematic parameters are the ones that don't fall easily into a set of existing real-world expectations. Either something is normal that shouldn't be, or something that we consider normal is abnormal.
4. Develop a multi-pronged strategy for how you as author will disseminate cultural knowledge
There are several ways to "get social and cultural information across." Use them all. If you use just one, I guarantee it will come off as weird, so try to balance the ways you get things across. Ask yourself:
a. which information will be evident in the setting?
For example, architecture says a lot about the history of a people. The presence of both classical stone buildings and apartment blocks of ultralight concrete implies a long history of technological development as well as respect for the legacy of the past.
b. which information will be integral to character behavior and judgment?
Manners and politeness will show up in dialogue and character judgments. It will show up in where the person goes, and how (where do they access transportation, for example?). It will show up in whether they notice "that person isn't where they should be," or "that person's clothes really mean he's showing off," etc.
c. which information will be taught?
Teaching should not be done by the author to the reader (unless you have an explicitly storytelling/teaching narrator). If something needs to be taught, it should show up in a natural teaching context within the society: teacher to student, adult to child, or insider to outsider, etc. There may be fixed methods (curriculum) by which such teaching is accomplished. I have a caste whose members, when in doubt, recite lessons to themselves. I don't do this with everyone, but it fits with the educational style of this particular caste group.
In each of these cases, the question of normal and abnormal is absolutely critical - normal and abnormal as defined by the point-of-view character(s). Any cultural detail that you explicitly describe will come across as "marked," or not entirely normal. If it's stuff that your people actually consider to be unusual, then fine. If it's stuff they consider normal, then you have a problem.
5. Make sure that the normal is defined by lack of attention, rather than attention.
To define something by lack of attention, you have to deliberately redirect attention onto something else. That something else can be a conflict between characters that causes them to say things to one another that they already know. It can be a "secondary detail," or some related characteristic within the normal parameter that has particular meaning - such as a hairstyle on a dark-haired head, if everyone has dark hair. It can be avoidance behaviors - say, when people of a lower class deliberately avoid particular types of interaction with members of a higher class in order to avoid unpleasantness.
Note: I constantly - and I mean constantly - see abuse and discrimination of oppressed groups indicated by direct insults or by direct conflict between the groups. Try to avoid this unless the presence of this conflict is the inciting event of your story for some reason. Oppressed people go very, very far out of their way to avoid conflict of this nature. You will be doing yourself a huge favor if you show the possibility of this conflict in avoidance behaviors and the characters' internal fears, and only show direct conflict in emergency situations. Because this stuff only happens when the people concerned are unable to avoid it.
6. Remember to defeat real-world expectations deliberately.
We have all kinds of "sets" that naturally go together in our expectations. "If there's a TV, there must be a phone" is one example. "If people are having intimate relations outside of an existing monogamous relationship, that must be bad," is another. The one I run into is, "If there are nobles living in a caste system, it has to be medieval." As author, you have to defuse these contexts deliberately. Show the different path technology took in your world. Or have characters casually discussing what would be taboo behavior for us. Or make sure to put electric lights on the first page just to say to the reader, "This is not medieval! SEE?" Just leaving it in the background is not enough, because our expectations are very, very strong. Maybe you've seen that internet meme with the message where only the first and last letters of each word are in the correct places, and everything in the middle is mixed up? And you can still read it? That is because the strength of your previous knowledge and expectations will be enough to build the word just on the basis of first and last letter (and maybe a hint of middle content). Believe me, only one or two details will cause an entire set of technologies, or morals, etc. to come into play. If you don't want them in play, you have to slap them down on purpose, as early as possible.
I could go into greater detail, but I hope this gives you a basic framework to work from. Good luck with creating your fictional societies!