Today's question asks: Tell us about the various myths about phoenixes that is around the world! Actually, this is a trick question (so don't think you got us, question-asker): there is only one Phoenix, which is a Greek mythological bird with particular qualities representing life and rebirth. But I think what you're really asking here is about various other mythological birds that are similar to or related to the Greek Phoenix, and we can do that, too!
The Phoenix of Greek mythology is a borrowed creature, but we're not entirely certain when it was borrowed or from whom. Herodotus first mentions it in his histories and claims that he learned about it in Heliopolis (Junu) from the Egyptian people who lived there in around 5 B.C.E., but Hesiod also mentions a few centuries earlier, when he uses its long lifespan to illustrate that the gods live longer still. It really became popular around the first centuries B.C.E. and A.C.E., after which it became a favorite symbolic creature in Greek and Roman mythology and eventually a well-known symbol in medieval texts.
The earliest descriptions of the Phoenix stress its great age; Hesiod, describing how long the nymphai can live, claims that the Phoenix lives three times as long as a raven, which in turn lives three times as long as a stag, which in turn lives four times as long as a crow (and then goes on to say that since the nymphai live ten times as long as a phoenix, they are long-lived indeed). Herodotus sets the bird's age at five hundred years old, claiming that every half millennium it returns to its nesting place to die, and that lifespan stuck and was repeated by various other Greek and Roman scholars all the way down to Ovid. Herodotus is the first to actually describe the bird other than its age, and he says it has red and gold plumage and resembles an eagle, and that it creates a beautiful funeral egg out of myrrh in which to encase its forbear, which it then carries to Egypt and leaves there to be burned (a feat he found particularly impressive, since he believed it flew all the way from Arabia in order to do so).
Later writers exaggerate the Phoenix's virtues even more, claiming that it is incredibly wise and understands all mathematics, that it is made of gold or shines with the golden rays of the sun, that it consumes only rare spices and fragrances such as frankincense, that it can never become sick and doesn't need to eat or drink to stay alive, that it is surrounded by a halo of fire, and even that it lives as long as the gods themselves (which of course means forever). The tale of the Phoenix's death and resurrection is also embellished and refined across the centuries by these writers, until it becomes the familiar version that became popular in the Middle Ages in Europe: the Phoenix lives for five hundred years, then returns to the nest where it was born, where it sings itself a funeral song and burns alive, only to be reborn anew as soon as the flames cool.
The point of the Phoenix's mythology is to use it as an allegory for the sun specifically, and later for light, life, and warmth in general. The Phoenix's journey from east to west (Arabia to Egypt) before it dies and then arises again mirrors the sun's journey from east to west each day, and its "death" in the evening only to return at the next dawn. The light, gold, and fire connotations it often has in various versions are linked to the sun originally, although they later become regarded as simply attributes of the bird itself as its mythology becomes more indistinct.
Obviously, the Phoenix is a pretty ingrained image in the western world - its origins and mythological imagery may not necessarily be common knowledge, but most people know the general idea of the immortal bird, and we name all kinds of businesses and brands and even cities after it, as well as having colloquial phrases based on its myth as normal parts of our speech ("from the ashes" is the most common, meaning something that resurges or returns after a disaster or setback). In fact, it's so ingrained that we tend to think of other mythological birds around the world as "types of Phoenixes", like this question-asker did, even though many of them probably have nothing to do with the Greek Phoenix. Which brings me onward to these other birds!
The Phoenix's closest cousin is the bn bird, better known to modern people as the Benu or Bennu. The Bennu is an Egyptian mythological figure, the embodiment of the ba (or personality-soul) of the sun-god Ra, which was said to be self-created and therefore an assistant to the major gods in their own acts of creating the world. It was considered immortal and often associated with Osiris, who was similarly deathless thanks to being resurrected as the god of death through his wife Isis' intervention, and sometimes appears in Egyptian art wearing Osiris' white crown. Unlike the Phoenix, which was described as a fanciful gold-and-red embodied sunset, the Bennu usually appears in Egyptian imagery as a heron, possibly because of a linguistic pun - bn seems to have had several potential meanings, one of them being "heron".
The Bennu is definitely related to the sun, as Ra's ba, and similarly eternal and undying, so it's understandable that it is often related to or outright conflated with the Phoenix, especially during the periods of history when Greek and Roman influence was strong in Egypt. We don't have any Egyptian sources that suggest that the Bennu ever dies, to resurrect itself or otherwise, but later Greek writers did recognize its similarities to the Phoenix and decide that this must be the Egyptian version of the same bird, and therefore wrote about the Bennu dying in flame on their own regardless of what it was doing in Egyptian myth. There's a healthy scholarly debate over whether the Phoenix influenced the Bennu or vice versa, thanks to Herodotus' writings about learning about the Phoenix in Egypt; some scholars think that he simply fancified or misunderstood Egyptians who were trying to describe the Bennu to him and thought they meant the Phoenix of his homeland's myths, while others insist that while the Bennu might originally have been more important as Ra's ba, later Greek influence over Egyptian religions functionally merged it with the Phoenix and its myths should be considered the same.
Of course, who did what and influenced who is an eternal Phoenix-and-egg situation in Greece vs. Rome vs. Egypt, so no one should really be too surprised.
To range a little farther north and east, there is also a fantastic immortal bird in the myths of Persia, the mighty and most holy Simurgh. The Simurgh is definitely a bird, and is in fact referred to as the mother or ruler of all birds, but she also has features of other animals, including often the head and/or paws of a dog, and sometimes the claws of a lion. The Simurgh is as eternal as the other birds on this list - she has supposedly outlived the universe itself several times, living through its birth and eventual destruction, and as a result is considered the wisest being in existence (outside of Ahura Mazda, of course, who she often appears as a symbol of). The Simurgh's immense age and wisdom mean that she knows the answers to all secrets and the cures to all ills, and she also guards the Tree of Life, which contains the seeds of all plants in existence and can create elixirs that heal any wound or sickness, so Persian heroes often set out to seek her to ask for advice or beg for some of the tree's seeds. Of course, they usually aren't up to the task of finding the Tree of Life and managing to impress the oldest non-god being in existence into giving them what they want, but that's heroing for you.
The Simurgh is generally considered to be much larger than the Phoenix or Bennu, capable of carrying off an elephant to snack on if she feels like it, and is usually described as being covered in beautiful bronze or gold feathers that reflect the colors of the world around her. Unlike the previous two birds, she's not really about the sun, though; she's a symbol of wisdom and purity, and is considered to purify the world and to protect it against the evil influence of the daeva and their minions. She's still strongly tied to a god as his representative, however, in this case the all-powerful Ahura Mazda, and she sometimes appears to let people know that he has sanctioned a particular place or act, or is used as a symbol of the god himself when depicted in art.
There are a very few mythic mentions of the Simurgh dying in flame, but they tend to be later ones and have never been incorporated into the wider body of myth about the Simurgh, so it's likely that they represent Greek influence filtering over from the Phoenix to occasionally color the Simurgh's myths. The Slavic version of the Simurgh that appears in Slavic areas that bordered on the Persian empire, the Simargl, has completely lost any such connotations, and is representative only of justice and protection.
Speaking of Slavic myths, though! The Zhar-Ptitsa, commonly referred to in English as simply the Firebird, is a Russian mythological creature that is often related to the Phoenix and other magical birds. It primarily appears in folktales, fairy tales, and heroic quests, where it is described as a bird with glowing eyes and fiery plumage, which brings great power but also generally great problems to anyone who is able to capture and keep it.
The Zhar-Ptitsa appears in many Russian fairy tales, but it's most famous in the modern era for being the subject of Stravinsky's incredibly popular ballet named after it, which retells the story of the heroic Prince Ivan and his quest to marry the daughters of the terrible Koschei the Deathless, a feat he attempts to accomplish with the help of the Zhar-Ptitsa, which he has caught. In this particular story, there are no negative consequences for Ivan from capturing the firebird, which helps him avoid Koschei's attempts to kill or confuse him and eventually gives him the secret to defeat him, but most often control of the Zhar-Ptitsa is a mixed blessing; Ivan Tsarevich is killed and dismembered by his brothers who covet the bird in one tale, and the wicked king who demanded the Zhar-Ptitsa be captured for him was later tricked into boiling himself alive once he owned it. Some stories caution against even picking up the firebird's discarded feathers, which are beautiful and alluring, but will almost surely bring trouble along with them.
(Incidentally, you can watch full versions of Stravinsky's The Firebird online, so go get you some ballet if that's your thing!)
Finally (at least for today!), we have the fenghuang, the immortal bird of Chinese mythology. Resplendent and multi-colored with connections to various Chinese astrology practices and imperial symbols, the fenghuang is a creature of symbolism, combining in itself male and female (literally - early forms of the bird during the Han dynasty had a male, the feng, and a female, the huang, but they were combined over time to become fenghuang, a single being), mercy and judgment (when it decides whether or not the current emperor is doing their job correctly but is also the imperial symbol of the empress and her kindness), and the connection between sky and earth. It's most often seen in imperial imagery and decoration (and is called the Ruler of Birds, similar to the Simurgh, as a result), but also commonly appears as a wedding symbol, since it represents harmony between men and women, and as a symbol of good luck on temples and at religious ceremonies.
The idea of the Phoenix as the template bird is so ingrained in western thought that the fenghuang is frequently referred to in the West as "the Chinese Phoenix", even though it has next to nothing to do with the Phoenix or anything it stands for. The fenghuang does occasionally have symbolic ties to the sun, and it is considered to be eternal and never-aging, but that's about as close as the two get.
Enthusiasts of the Four Celestial Beasts in Chinese mythology, incidentally, may also have heard about the Zhū Què, called the Vermillion Bird in English, which is one of the four guardian constellation creatures that dominate much of Chinese star lore. These two aren't actually the same bird, although they are usually represented similarly in art and therefore confusion occasionally arises between them, and the Zhū Què is even further from the Greek concept of the Phoenix than the fenghuang is.
There are more ancient and long-lived mythological birds out there, but I think that covers the most famous!