Elves are immortal, but there are still some loose stages to their life and distinct stages of early development.
Creation
Childhood
Elf mothers are with child for 1 year. Once born, elves' minds and bodies initially grow rapidly, advancing past the infant and toddler stage within the first year. Development then slows down and an elf is mentally in adolescence for roughly the next 49 years. The body develops more slowly than the mind, thus elven children seem to be older than they look—or smaller for their maturity and intelligence. This is why other peoples often confuse elf children and sprites.
Adulthood
Once they are physically mature at 100, elves do not age any further. They retain the appearance of a very healthy and well-formed 20 to 30 year old human, though have a more youthful complexion and air about them. Over time elves can "age" emotionally into a sort of weariness or ennui. This state is also called ingem, lit. 'year sickly', a term used to refer to an oldness of attitude casued by seeing many years and not an oldness of body. This state can put an elf in danger of gwanu.
Elves do not take long after puberty to find mates, who they pair with for life. Parents are often heavily involved in the process and arrangement of matches can take place decades in advance of puberty. This gives parents and children a long time to resolve issues and come to an agreement. When the time agreed upon arrives, elves bind themselves to one another. They exchange jewels to wear as symbol of their binding, usually made the most skilled family member, and sometimes there is a feast. Many elves have very private exchanges of words and jewels and that is acceptable for binding.
Fertility
It is not common for elves to have large families as they have much lower fertility rates than humans. Fertility lasts until age 3,000. A sexual union does not normally result in impregnation, but requires a strong act of will and focus of the mind for that union to be fertile. Some say then that elves are only born of magic, given the will that is required to bring them forth. The sexual act is powerfully bonding for elves, so much so that it is compared to a carnal marriage of sorts, and they experience a greater grief and pull than humans when absent from a partner. Elven sex exhibits an extended response cycle, lasting longer than human copulation, as pleasure, not procreation, is the primary purpose. Sex is another of the many arts and pastimes that elves relish in their infinite but becalmed revelry. Elves also experience little to no refractory period. When a child is conceived, though, there is a refractory period of nearly a week for both man and woman.
Old Age
Elves are considered old at around 9,000. It is at this point that male elves begin to show signs of aging with the appearance of a beard. At first the beard is normal and "unaged" as the beard of a young adult human or dwarf. By the age of 10,000 their beards can begin to weaken and become scraggly. Elves will then start to show the first physical signs of aging, but these will remain at this stage for the rest of their life. It is said that there are elves that still live who have been since the beginning of their folk or even since the beginning of the world, e.g. the Arnaden.
Death and Rebirth
Elves are not taken by sickness or disease. Starvation will cause them to go into a torpor until they are supplied with nourishment. They can be killed by wounds, however, they have a much greater resilience than humans. The same goes for emotional distress, but there is a backdoor by which intense anxiety, depression, and other intense psychologically stressful conditions can induce death. The heaviness of the spirit then prevents it from returning. Banishment may have the same effect.
When elves die, their bodies are separated from their spirits as all people are. Unless they are ghost-bound, their spirit wanders to the place that is most appropriate for them and it there in that significance that they reform their bodies at the age that they died. A god (the chief deity of elves or an invested tutelar) may weigh in and judge that there be a time of waiting for that elf spirit—for reflection, repentance, and/or spiritual refinement—or, it they are a significant threat, that the elf be bound from returning to bodily form. The spirit will endure and can manifest, though, as the spirits of elves cannot sleep like mortals. Elves who have done great misdeeds fear death because of the purification they will face in the spirit before they are allowed to return. If cut down in their prime, an elf is quietly returned without any trial, but the spirit trials of some elves have been grand affairs. There have been times when it is judged for them to return in a different body, as punishment or disguise. Unless they have a fate with unwound threads, a slain elf will continue on in a typical cyle of reincarnation.
[An elf that dies from a broken heart will break from the circle of reincarnation, thus the Fate of Elves is that they are in a cycle of enduring death and the horrors of the world until it breaks them.—No, this would be the fate of half elves or any with "strong" elven blood still in them (1/2, 1/4, 1/8; ?1/16, ?1/32). 29 or 30 generations = all possible ancestors (causal chain).]
[An elf that dies by grief will leave their body behind unfading, not decaying, as if in enchanted sleep. Protecting the body or some basic tending is necessary as it could be kidnapped or eaten by beasts. When their grief is resolved, their spirit will return to the unmarred body. If the body is in fact harmed to the point of a usual death, then the grieving elf will be reincarnated [instantly or by trial?] and they will be pleasantly surprised to be alive and experiencing bodily sensation again or they will continue in a cycle of grief death.]
Elves are always forced to come back. That is their fate.
Sources
Elven Life Cycle on Tolkien Gateway
Death in Tolkien's Legendarium
Elvish Philosophy of Death and Incarnation
https://middle-earth.xenite.org/where-did-the-power-in-the-minor-rings-come-from/
Elven reincarnation citataions
Why do elves remain in Middle Earth? on Sci-fi Stack Exchange