I have read in some philosophy forums that Plato believed that death was not the end and he most likely believed in reincarnation.
Are there any texts that confirm that Plato actually believed in reincarnation?
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Yes, and the Phaedo and the Republic (to cite just two texts) provide evidence for this.
Running through the dialogue has been the thought that soul and body are sharply distinct and opposed. In fact Plato notoriously wavers, between different arguments, in his treatment of the soul/body relation and the nature of the soul; but in Socrates' extended discussions of the philosoph- er's attitude to the soul and the body (64a-69e) and their respective affinities (78b-84b) he develops the idea that excellence for a soul lies in separation from the body and defectiveness in attachment to the body and commitment to its concerns. In this context he describes the good soul as parting easily from the body at death, while the bad soul lingers on round it, and because of its desire for embodiment is compelled to re-enter other bodies again, of a kind appropriate to its former life (80d5-82c8). Reincarnation thus appears as a punishment for a bad life, and the highest k>ind of virtue is said to belong to the philosopher, who by refusing to identify with the body's concerns renders his soul at death "pure", unattracted by the body and presumably not liable to reincarnation.
It is true that in the only passage where reincarnation explicitly figures the treatment of it is ironical; various types of people are reincarnated as appropriate animals. But this is no reason to minimize Plato's commitment to the idea. He is deadly serious in his repeated insistence that the best fate for a soul is permanent escape from the body, and that being returned to a body is appropriate as a punishment. He is ironical in his sketch of the fates of various kinds of people; but he refers back to this passage in all apparent seriousness later in the myth (108a8), something which indicates that for him reincarnation is not a crackpot personal belief but an appro- priate expression of important truths about the relation of soul and body in the individual person. (Julia Annas, 'Plato's Myths of Judgement', Phronesis, Vol. 27, No. 2 (1982), pp. 119-143 : 126-7.)
☛ Myth of Er
After death there is still a sentence pronounced by divine judges, which sends the good to heaven and the bad to punishment (614c2-615c1); there is still a distinction between the curably wicked and the incurable, who are sent forever to Tartarus (615c5-616a7); in all these myths the judgement takes place at a crossroads, where the good and the bad, who have come along the same path, are parted to go to their separate destinations.
But the big and immediate difference in the Republic is that there is no longer any suggestion that this is a final judgement. What death reveals to Er is that he and everybody else is on a cycle of birth and rebirth (617d6-7). There are now roads coming back from heaven and hell, as prominent as the roads going there (614c1-4). The judges sit in the middle of the open- ings to these four roads, and there is a constant stream of souls going up or down after their rewards and punishments; this is a constantly ongoing judgement, with none of the finality attaching to the decisions of the solemn judges in the Gorgias. The siting of the judgement between the four openings at the somewhat crowded junction where souls pass into and out of lives brings home bluntly the fact that the judgement has shrunk in importance, being now only one episode in a continuing cycle of endless reincarnations.
The rewards and punishments are still to be reckoned with; one pays tenfold for one's crimes (615a4-cl). So, since anyone has reason to avoid tenfold punishment for what one has done, they still have some significance for the individual. But their significance has been changed by the way that the judgement appears, no longer as a final judgement on my life, but as merely an episode at the end of my life which is part of a cosmic cycle of reincarnations. For there are two very major ways in which my attitude to rewards after death for being just in this life is likely to be affected by the knowledge that Er gains, that my life is only one interlude for a soul on the wheel of reincarnations. (Annas, 131-2.)
Endnote on Platonic myth
Plato's talk of reincarnation often occurs in the context of a myth. This is no ground for dismissal of seriousness.
Plato's myths are often ignored or downgraded because it is thought that he takes all myths, including his own, to be mere mythoi or stories, which are all to be despised by contrast with logoi or rational discourse and argument. This is, however, too simple. Mythos and cognate words originally mean no more than "speech", and the usage survives in Plato whereby mythoi and logoi are put together and both are opposed to action (e.g. Republic 376d9-10). By Plato's time mythos has come to mean some- thing like "story"; to favour mythoi over logoi is to favour storytelling over argument. Given his stress on the importance of reason in our lives, it is not surprising that we can often find Plato displaying a general hostility to stories, and it is not hard to find passages where he abuses or despises (mere) stories as trivial, suitable only for children or lightweight enter- tainment.6 He is especially hostile to the stories that we think of as traditional "Greek myths"; Republic books 2 and 3 attack them as immoral and misleading, and he insists that they should not be allegorized or explained in terms of physical theory; he refuses to find rational depth in them.
But this hostility or indifference concerns the content of particular stories. Plato nowhere says or implies that there is a single all-purpose distinction between storytelling and reasoning such that all stories are necessarily stupid or immoral. He in fact clearly believes that some mythoi, stories, do have rational depth. (Annas, 120-1.)
This is the word of Lachesis [...] : Souls that live for a day, now is the beginning of another cycle of mortal generation where birth is the beacon of death. No divinity shall cast lots for you, but you shall choose your own deity. [...] And after this again the prophet placed the patterns of lives before them on the ground, far more numerous than the assembly. They were of every variety, for there were lives of all kinds of animals and all sorts of human lives.
Yet if at each return to the life of this world a man loved wisdom sanely, and the lot of his choice did not fall out among the last, we may venture to affirm, from what was reported thence, that not only will he be happy here but that the path of his journey thither and the return to this world will not be underground and rough but smooth and through the heavens. [...] He saw the soul of Thamyras choosing the life of a nightingale; and he saw a swan changing to the choice of the life of man, and similarly other musical animals. The soul that drew the twentieth lot chose the life of a lion; it was the soul of Ajax, the son of Telamon, which, because it remembered the adjudication of the arms of Achilles, was unwilling to become a man. The next, the soul of Agamemnon, likewise from hatred of the human race because of its sufferings, substituted the life of an eagle.
Some version of reincarnation - which was seen as a Pythagorean belief - can be found in other dialogues. Both the Phaedo and the Gorgias conclude with myths of afetrlife judgment, and the Phaedrus (246b–256e) speaks of the reincarnational cycle.
But we have to consider the metaphorical structure of the Republic regarding death and rebirth : especially birth out of a cave or some other underground place (see also the Allegory of the cave, 514a).
See also : Stephen Halliwell, The Life-and-Death Journey of the Soul : Interpreting the Myth of Er, Ch.16 of G.R.F. Ferrari (editor), The Cambridge Companion to Plato's Republic, Cambridge University Press (2007).
Plato said, “Apply you both now and in the next life. Without effort, you cannot be prosperous. Though the land be good, you cannot have an abundant crop without cultivation.” Thus he personally believed in metempsychosis.
I outline several arguments against taking Plato literally concerning reincarnation here:
Arguments include (1) Plato's general reluctance to merely speculate (as opposed relating philosophical truths one may verify by contemplative experience); (2) his allusions to human-to-animal transmigration -- which is both a priori implausible and something many ancient philosophers expressly reject; and (3) the parallel between progressively lower reincarnations in his myths and the "Tyrant's progress" section of the Republic, which posits a succession of progressively worse states of mind. The most important argument against literalism here, however, is this: understood allegorically, Plato's discussions about reincarnation supply an extremely insightful and practically valuable psychological model for progress/retrogression in personal moral development. So we must ask: is it more likely Plato is using a helpful allegory to address one of the most significant issues we face in our moral life (maintaining our mind on a virtuous level, and avoiding its 'descent'), or that he's relating old wives' tales?