The answer is "yes." You do all of the above. Each system serves different purposes.
The first big division to focus on is laymans' names versus scientific names. You will find that a realistic community will have both, because they serve different purposes. Laymans names are more focused on practicality. Two things that are obviously different will have different names. Two evolutionary similar things with different uses may have very different names. They are easy to remember and easy to teach. Scientific terminology, on the other hand is designed to support science's key tenants: repeatability and reproducibility. When you write a paper and someone tries to reproduce it, it's important that they can identify the exact same species.
As a fun example, consider two stinging beasties that we deal with as divers in the ocean. The first is the hydroid:
These little buggers hang out on rocks. If there's nasty weather, they break off and the little pieces can sting you. The next baddie is the jellyfish:
We all recognize a jellyfish when we see them. Some are extremely poisonous, others like the moon jelly above are more cute than anything. Now these are layman's names. They're easily identifiable, and practical. The scientific names may be less practical. The first picture is of Turritopsis nutricula, in the class of Hydrozoa. The latter is Aurelia aurita, of the class Scyphozoa (a class that is typically known as "true jellyfish").
Now what about this guy?
He looks an awful lot like a moon jelly. In fact, his layman's name is the Immortal Jellyfish, for its unique ability to return to its larval stage if needed. Scientists, however, call him Turritopsis nutricula. If this doesn't sound familiar, look up a bit. It's the name of the spiky looking hydroid I showed a picture of earlier. These are actually two pictures of the same species, in different parts of its lifecycle. The former was a picture during its hydroid stage, rooted on the ground. The hydroids then emit tiny "medusae," which are the jelly-fish like things we know and love. The scientific name is far more concerned with a precise name for the species than it is the convenient identification at a glance.
Now not only do we have two pictures of things which may be given different layman's names but the same scientific name, but we also mentioned the different classes. The idea of classes is relatively new. It only started to appear in the 19th century, and only gained fame as Darwin promoted it. The idea that everything is connected evolutionary into a phylogenetic tree is quite a new concept as far as science is concerned.
Which points out the next piece of the puzzle for you. Eventually you will want to put a new level at the root of our phylogenetic trees, indicating the different origin planets. This is mentioned in other answers, but here we can see why. The explicit purpose of these trees is to show a common genetic heritage. There will be no common genetic heritage between Earth and your planet 14LYr away (or will there?). Thus, you should not borrow from any of the existing branches of our tree. Branch off lower, and start a new one.
However, that will come later. It took a long time to figure out all of the different branches of our tree of life. And, honestly, it's not that helpful to anyone outside of a biologist who is actively researching deep evolutionary history. You would probably start with just a genus and species (such as Turritopsis nutricula or Homo sapiens), and start cataloguing that obvious part of the tree. Later, as you learn more about the planet and it's history, you'll be in a better position to solidify the lower branches of the tree with the evolutionary history. But, in the mean time, genus and species is so effective that we typically refer to all living species by their genus and species (it's also called a "binomial name"). Sometimes we even abbreviate the genus (such as "H. sapiens") because there's enough information there to provide all the scientific precision required.
But while you're giving things scientific names, you will find they get layman's names as well. My garden has Solanum lycopersicum, Daucus carota, and Spinacia oleracea, but if you were to ask me what's in my garden in a conversational setting, I'd tell you "tomatoes, carrots, and spinach." You can see why the laman's names have a use!
