My philosophy teacher says that Plato was influenced by Hindu and Buddhist teachings, noting similarities such as belief in reincarnation, and returning to the form of the good and moksha.
However, this seems unlikely to me that Plato would have heard about Hinduism, because to my understanding India was a relatively alien place when Alexander arrived, years after Plato's death.
Is it possible that cultural and philosophical transmission through trade routes such as the silk road influenced Platonism?
However, this seems unlikely to me that Plato would have heard about Hinduism, because to my understanding India was a relatively alien place when Alexander arrived, years after Plato's death.
That is an eminently valid objection, the more so given that Plato's understanding of reincarnation came from several earlier figures dating back to the 6th century BCE, and was particularly influenced by Pythagoras. The idea of Indian influence relies heavily on a "well, we can't rule this out with absolute certainty..." kind of argument. It can be taken seriously, but it's very much doubted. Some bibliography cited by the New Pauly article on the subject:
The basic idea is that while it's not strictly speaking impossible that there was a connection between reincarnation as taught by several early Greek philosopher-mystics and reincarnation as taught in India, there's no actual evidence to connect them.
Now, while the evidence in favour is essentially non-existent, the obstacles to the idea are moderately substantial:
People who were contemporary with, or shortly after, the Greek mystics and philosophers who taught reincarnation (Pherekydes of Syros, Epimenides, Pythagoras, Hermotimos of Klazomenai, and Empedokles) aligned those figures with teachings from Thrace and Scythia, i.e. the north, rather than the east. (Herodotos 2.133 claims, falsely, that reincarnation was a feature of Egyptian religion; he was probably thinking of Pythagoras, and conflated features of separate, unrelated cults.)
In Greek thought, reincarnation was the "transference of the soul", or metempsychosis. Buddhist thought, by contrast, does not necessarily incorporate the idea of a soul, but understands the continuity from one life to the next in a much more abstract sense.
There's no such thing as karma in Greek thought. By contrast, in Buddhism, it's pretty fundamental to reincarnation.
Another case of independent origin of the idea: Caesar reports on belief in reincarnation in the Druidic religion (BG 6.14.5); though Diodoros of Sicily connects that with Pythagoreanism, very implausibly (5.28.6).
(Incidentally, if you do more research you may come across the alternative Greek term palingenesia, literally "being born again". This is not reincarnation in any popular modern sense; the Greek word for reincarnation (in the popular sense) is specifically metempsychosis or metensomatosis. Palingenesia refers to rebirth into the same life you've just had: it's a neo-Platonic idea that is nowadays usually associated with Nietzsche, who plagiarised it for his "Myth of the Eternal Return".)