Animals
When animals evolve, they typically reach a measure of sentience after two or three evolutions. After attaining sentience, they evolve towards a more "Humanoid" form, at which point they can be counted as one of the Races. However, this is typically achieved over generations, instead of by an individual.
When an animal manages to evolve, it is individually elevated, but progress towards higher evolutions is rare. So typically they need to pass on their evolved state by breeding with another evolved individual. When an evolved animal reproduces with another individual of the less evolved species, the children will almost always be in the lower state, but can rarely be born in the evolved state.
An animal that lived as the base form, then evolved will typically have higher stats than one born at the higher evolution. But having a larger population of the evolved species will give more opportunities for growth.
Predatory animals have a higher tendency to undergo Mutation rather than simple evolution when compared to non-predatory animals. Simply because a strong enough predatory animal can hunt and consume weak monsters, thus taking in mana and eventually causing a Mutation. Though it is still possible for herbivores to mutate.
Monstrous versions of Animals are always stronger than the non-monster varieties, thus the Races descended from them are also relatively superior. For instance, Dire Wolf Beastmen are much stronger than Wolf Beastmen, even when both were born in such states.
Insects/Arachnids/Etc
The evolution of Insects and the like is unique in that it trends towards Collective Intelligence. In other words, they become Eusocial Telepathic Hives. Then they build intelligence by increasing the size of the hive, rather than trying for higher evolutions. Though evolving can certainly speed them up. Typically, they count as a Hive Mind Race at ten thousand individuals and two evolutions after reaching that Eusocial status.
Their stats even pool into their groups, allowing even the smallest of insects to end up with some points in the various stats instead of being too weak to even register.
Insects that are naturally Eusocial have a significant advantage. They still need to evolve, but a single evolution and a large enough population will cause a group to reach the Status of a Race.
Monstrous versions of Insects actually grow to be larger and more powerful, and thus evolve towards sentience as individuals. Though if they are Eusocial before becoming monsters they will still work together in groups, and likely gain Telepathy.
Monsters
Monsters, like other creatures trend towards greater intelligence as they evolve. Uniquely though, innately stronger monsters require more evolutions to reach the status of a Race. A slime or goblin would evolve into a Race after just two evolutions, but a lesser dragon would require four.
They also sometimes take on the evolutionary traits of the other categories of creatures, depending on which they evolved from.
Plants
Plants do not go through evolution in quite the same manner as other creatures. Instead their evolution state, and thus progress towards status as a member of the Races is determined by their growth instead of their level. And the evolutions themselves don't actually increase their stats. Like insects they sometimes develop collective intelligence, and often telepathy.
They are unable to pass their evolutionary state down to their offspring, thus only by the progeny's aging and growing can they increase the population of a given evolution. This remains true even when Plants reach the level of one of the Races.
Monstrous Plants retain the fact that their progeny will have to grow to evolve, instead of keeping the parent's state, but their offspring will at least still count as monsters, instead of being the base species.
Races
All Races possess the ability to evolve at least once. Though they can vary greatly in the circumstances of their evolutions. For instance, one race might simply evolve upon reaching their natural level cap. Where as another might have the option to evolve at level forty with a different stronger evolution available if they wait till they reach level one hundred. Yet others can have multi-staged evolutions. Evolving once when they reach level fifty, then again when they reach level one hundred in their evolved state, then a third time if they live long enough to have their level cap, and level, all the way up to two hundred.
There is also a Special Evolution that is only available to the Races. Subspecies Evolution, which takes two forms, Progenitor, and Metamorphic. The prospective Progenitor of a new Subspecies must complete a special challenge posed by their race's god. For instance, in the case of the Elves, growing an entire forest of no less than a hundred thousand trees through individual effort. Which has been accomplished over a dozen times throughout the Species' long history.
Upon growing a large enough a forest, an individual elf would be allowed to commune with the goddess Allelea. At which point they can request a few specific changes to their Race for the new Subspecies. Though most of the changes are up to the God, Goddess, or Deity's own whims.
Subspecies themselves will have their own separate challenges from the original race, thus sometimes leading to chains of Subspecies, which also has happened a few times with the Elves. Metamorphic Subspecies Evolution occurs when the Progenitor of a new Subspecies offers a member of the original species the chance to become a member of the subspecies, thus allowing a population base for the new Subspecies to be established.
The Races are always unable to pass on their evolved state to their children. The children of the Races will always be in the Race's, or Subspecies' as the case may be, most basic state.
Spirits/Elementals
The evolution of Spirits and Elementals is a very mysterious thing. What can be easily established is that Sprites tend to evolve into Spirits, and Elementals are closely related, but not necessarily an evolution of either of the other two.
What are automatons in this?
An interesting question. Technically speaking. None of the above. Automatons only count as living things when a soul is put into an Automaton body, otherwise it is more like an item. Because of this "Automatons" cannot evolve the same way as other races. The soul within the body however, can undergo most Special Evolutions. Of course, if the technology comprising them advances far enough, they will eventually count as living things even without the Animation step. At which point they would be treated as Monsters.