I'll take a few liberties with this question, namely:
- For the purposes of this question "race" will include species.
- I will not go into the detail of working the Chinese Zodiac.
- I'll assume we are talking about creatures that you may confuse with animals.
- I'll base my approach in engineering.
I'm not an expert, this is not a treaty. Consider other points of view aside from what I present here.
What I provide below could work as a framework to attach all sort of details about your race. Yet, you shouldn't be required, so, I'll be using the mark "[⌥]" hints, skippables, and optional developments.
If you wish to make an apple pie from scratch, you must first invent the universe. -- Carl Sagan
Go watch (or read, or listen in audiobook) Cosmos. Yes, the Neil deGrasse Tyson version is ok. But listen to Carl Sagan talking about hypothetical creature living in Jupiter. There is no better inspiration.
[⌥] Skip: if your universe and world is just like ours.
Whatever we decide to do with our creature project must be based on the conditions on the universe in which we are working. We should consider if we have tweaked the laws of physics, if we have added any exotic matter or unfamiliar particle, or if we have allowed any form of magic. Also, consider the situation of the world in which you plan to put the creature. Consider if it is a planet that orbits a star, is it a disc-world, a ring-world, a Dyson sphere… whatever. You have the requirement to create a creature that can survive there.
Remember, this is Worldbuilding.
Identify all the traits and constraints for the race you want to create. These are two distinct sets of requirements. For example you may start working with a known environment and wonder what kind of creature may live there; in that case you will have more constraint requirements to begin with. On the other hand you may try to create a creature that resembles some description; in that case you have more trait requirements.
The more requirements you have, the harder will be to design a viable live form. A good idea to mitigate this problem is to prioritize your requirements. For example if you start with A) "Humans can mount it, it can fly, and it looks like a horse" you will end up with something different than B) "It looks like a horse, it can fly, and humans can mount it".
As you may notice, the requirements interact with each other. In the example above A may be a giant bird with horse head, while B may be a flying horse that when you try to ride it then it can't fly - or may even die.
[⌥] Hint: Keep the number of requirements at the bare minimun for the race to work in your setting. You don't an extensive list of all requirements. In fact, if you find something missing you can always add it later.
Adding one more requirement will multiply the complexity of the design because you need to consider how that requirement interacts with all the others. So, again, have priorities, and be willing to drop a few in order to make a viable creature.
Another example: if you are making vampires, drinking blood is probably on the top of the list, and then there is the hard stuff that is likely to be drop out such as having no reflections or turning into a bat, and finally at the lowest priority any glitter stuff.
Note: before going into the design, I want to encourage to do research. Search in zoology, mythology, and crypto-zoology, and of course sci-fi and fantasy for things similar to what you want. It will save you time later.
When designing the live form, you need to consider some extra information that you may not be requirements: What is the origin, and what is the environment in which it lives.
For example, you may conclude that the creature you want could only be possible if it is genetically engineered, or requires cybernetic enhancements, or could only evolve in a planet very different from Earth. And that could or not be a requirement.
Of course there are two main kinds of design:
Biological evolution: If the creature evolved, you need to justify each of the traits in one of these ways: 1) how does it help the creature survive? 2) Why sexual partners may prefer the feature? 3) What other creatures have similar traits? (In particular if you can find examples of real creatures).
Artificial construct: If the creature is artificial then it may depend on the race of the creators for survival. In this case the question to answer about the traits is why do the creators opt for this trait? (Answer could be practical, but could also be economical).
[⌥] Optional: With evolution, if you want to explain how the traits developed (to invent fosil records or stuff like that) you need to consider that any trait you add must have some use in its minimal form. If you like "it is a frog that by mutation had tiny wings" those tiny wings would have to have some benefit for the frog because otherwise evelution would very likely select it out.
Of course, that's a daunting task, below I present an approach to construct a creature from the ground up.
[⌥] Hint: how strict you are on this is bound to the target sci-fi hardness you want. As per convincing the audience, propinquity and repetition helps to make people think that something is normal. In particular if they know the stuff from other fictional work. A shortcut may be to have your characters react as if it were normal. Contrast with making a monster.
Decide: What is the medium in which it thrives? What substances are common? How strong gravity? What is the temperature there? What is the medium pressure? What traits does the creature need to survive there? [⌥] Optional: How strong is magic there (or the force, ki, chi, chakra, cosmos, ether, higgs field or whatever)?
Decide: Does the creature float there, or is stuck to a surface? Note: I'm talking about natural buoyancy, not locomotion.
Decide: How big is the creature? How does the body support itself? a skeleton, an exoskeleton, a system of fluid-filled bladders...?
Decide: How does the creature resist or take advange of the conditions of that enviroment? [⌥] Hint: rough stokes, no need for details.
Describe the body structure as a tree (starting from the part that has the heart, also starting form the part that has the brain, and from the part that has any other organ that needs to reach the whole body, if any). Does it have bilateral symmetry, radial symmetry, other? Are there counterparts in real life? [⌥] Shortcut: describe it like a known animal, perhaps point a few differences.
Solve its locomotion. Does it walk, run, gallop, trot, swim, fly, float, glide, soar, jump, attaches to another creature, teleport...? What are the requirements for that locomotion to work in the environment you did choose? [⌥] Shortcut: if the description is close to a known species, and there are no major enviromental or mass changes, you can probably handwave this.
Note: consider also other weird stuff like split body, and multiple hearts, etc. Any unique trait, make sure to make note of it.
Up to here you have a viable creature, but we want a race. Remember the genetic imperative, live is about three things: Feeding, Reproducing, and Not-Dying.
Solve the nutrition. How does the creatures get nutrients? From what or where do they get the nutrients? Do they even need nutrients? [⌥] Hint: they probably do. Also, you are not required to describe a food chain (e.g. "they eat fish" is ok).
Solve the reproduction. Do they reproduce? Is it asexual? If it is sexual, how does the biological sex determination works? Is there any sex differentiation? [⌥] Optional: How do the individuals find partners? What kind of care is given to the offspring? Does the offspring kill and eat the parents? [⌥] Hint: there is no need to come up with unique scheme for sex determination, just say it is chromosomes.
Solve the defense mechanisms. How do they defend tehmselves from predators (if any)? How do they fight? What may the creature do to appear more threatening? [⌥] Hint: rough stokes, no need for details.
[⌥] Optional: Decide how different are the individuals of the race, do they all look pretty much the same? are there distintive (sub)races? is it all crazy like dog breeds?
You did all that? Good. Go over the list again; make sure it is all consistent. Once you are done, you have a race.
Their society, their biology, their history, everything.
This question is too broad, and primary opinion based, and has a bounty - at the time of writing.
First off, a suggestion: go study history of science. Not big history, not universal history, not the history of your country. Instead, study the history of the scientific discoveries and the technological advances of human kind. It is technology what shapes economy and behavior, and so, it is technology what ultimately can change culture – technology and the "natural" death of the people with outdates ideas, but technology nonetheless. Yes, Cosmos again.
Your cultural design starts with means of communication. Do they use sounds, signs, telepathy, something else? [⌥] Optional: This is also where you want to start if you need to create a language, a naming scheme, family names, etc. You don’t need to develop a complex language to move on, but if you can specify grammar and vocabulary it will add a huge depth and realism to your culture.
Once you have some basic idea of communication, you need to move to technology. Think about what resources do they have, what kind of tools they could make, what kind of environmental challenges they have that could be better addressed with technology. This will give you their technological focus and also a rough idea of where to put manual labor. [⌥] Optional: Again, you don’t need to develop a full technology tree, doing so is daunting itself (I have tried) but any approach to the technological ages you can describe will add even more depth your culture.
Note: you may want to dedicate some time to developing the world, because that will tell you about geographical features, resource deposits, and the habitat of other life forms with which your culture may interact with (for hunting, taming, domestication, etc.)
Next up, is economy. This is bound to resource abundance and scarcity; and also to the needs of the creatures. Once the population is large enough, the production of the vital resources will have technological improvements that allow a distinct group of individual to provide the whole population, but why would they? Are they forced? Do they trade? Is there even private property? Do they a have common consciousness?
That gives the firsts strokes of social classes in your society.
[⌥] Optional: If you want to go deeper into the social structure, try to imagine how this kind of things could work there (if at all) – no particular order:
You may have decided how the society is on the current stage, but you are yet to say how it got there. To fill the gaps, you need the other approaches of history: Great Man Theory, and It-is-a-bunch-of-battles Theory.
Consider the different stages of technological development, for each one: What are the social problems in that technological period? Are there any internal or external conflicts? Who solved this? Who are the great generals/scientists/inventors/artists/engineers/merchants that moved history along to solve those social issues and to reach the next technological period?
[⌥] Hint: No, you don’t need to write about how they met or anything like that. All you want is a timeline, a series of events, that then you could explore in detail if needed.
[⌥] Optional: you may take that map of the region where your culture develops and start plotting how far it expands per technological period. This also helps to find how they contact other civilizations or discover new resources.
[⌥] Hint: Having problem with some part of history? Create an RPG campaign! Or ask for help online or whatever, not like we have a site on worldbuilding.
Everything else design
[⌥] Everything else is optional: You can explore the mysterious parts of their psyche: how diverse are their personalities? what are the moral values of the society? what do they consider wrong? do they dream? of electric sheeps? What do they consider beauty? What activities do they do for social bonding? Do they have a sense of humor? Can they catch metaphors that may attempt to fly over their heads?
[⌥] Hint: I guess what is left are the artistic expressions, fashion trends, and similar. You can always approach these as the development of your work needs them.