In my philosophy class, the professor told us about the allegory of the cave and how it relates to the Forms. From what I understood, regular people only see shadows of the Forms; but by doing philosophy, one can free himself from the chains and catch a glimpse of the Forms. Which Forms can philosophers catch a glimpse of?
"Which forms can philosophers catch a glimpse of?"
Especially those of the Good the True and the Beautiful (In Platonic philosophy, or traditional Political Philosophy). The whole discussion in Plato's Republic, so called in the translation, is on the form of Justice which is a part of the Good. It's basically the question: What is the best way to behave? The ordinary man adjusts to convention and vogue mindlessly, but the philosopher holds back, withdraws into death as it were, and from there surveys the idea of Justice attempting to find its highest annunciation, to determine it as a principle to guide one's deeds: Socrates thus says: The Just is what improves human beings. To follow this wouldn't suit most men, and, in fact, Thrasymachus grows enraged and frightens Socrates by turning beet red when he hears of this idea (provided we interpret this as anger and not a "blush" as is often supposed). So, he is is like the ones who would kill the man who comes back down from the sun of the idea proper, into the region where the shadow, the opinion that Justice is the "advantage of the stronger party" is regnant.
This by no means exhausts the interpretation of the passage, but I give an initial indication of the subject matter only.
If any specific Form is apprehended by the escaped prisoner it is the Forms of the Good, which Plato symbolises as the Sun at Rep. VI.508e-509a. So far as I can see, the text indicates no other particular Form. And - Plato is full of surprise - it may not indicate a Form at all. This is a point for later.
Here's a standard summary :
At Republic 514 a I Plato makes Socrates introduce the Cave as 'an illustration of our human nature in reference to education and want of education'. He tells us that the men chained in the Cave can see nothing but the shadows on the rock face in front of them which are cast by puppets being moved along the top of a low wall behind the prisoners, beyond which a fire is burning. Likewise they can hear no sound save the echo of voices uttered by the puppet operators behind this wall. Such prisoners are said to be homoious hemin ['like us] in being able to see only shadows of themselves and shadows of the phenomena. Yet, if someone were to turn such a prisoner round bodily and oblige him to look at the fire and the objects on the wall and compel him to recognize these things from their familiar shadows, the prisoner would still consider those shadows seen earlier to be truer than the objects now shown him.
In 516 a-d it is also suggested that such a prisoner would resent being released and dragged up into the daylight outside the cave where he had always been kept, and that he would temporarily be blinded by the unfamiliar light. But later his eyes will perceive shadows and reflections by day and the stars and moon by night. Then finally he will look up at the sun, and recognize it as the source of light causing the seasons and all the other phenomena he observes. (R. G. Tanner, 'ΔΙΑΝΟΙΑ [dianoia] and Plato's Cave', The Classical Quarterly, Vol. 20, No. 1 (May, 1970), pp. 81-91: 85.)
If, which is not clear, the stars and moon or other things itemised here refer to particular Forms there is absolutely no indication of which Forms they are.
The peculiarity of the 'Form of the Good'
The above account, which is standard, assumes that what we call 'the Form of the Good' really and truly is a Form. There are grounds for caution here.
The Forms are distinguished as having 'being and reality' - they are the 'objects of knowledge' - unlike sensible particulars and everything in the sense-based world. Yet the Form of the Good (he tou agathou idea), is beyond the being and reality (epekeina tês ousias) of the Forms (VI.508e2–3 & 508b8-10). A natural reading of this is that what we call 'the Form of the Good' is not itself a Form, since it transcends the Forms.
On this reading, the escaped prisoner does not apprehend any of the Forms but only the Sun which, symbolising the Good, is beyond them in being and reality.