I only know a bit of what I've looked up on my own, but perhaps I can get you started, although my approach is sure to be skewed as compared to most others, and I may make glaring errors. I don't know if you are deshi; if so, you may be able to correct some of my misunderstandings concerning traditional practices on your own. Also, I am giving a sort of bulletpoint approach to traditions that have existed for thousands of years. I am therefore dealing with some of the most straightforward and systematic ideas, which may not be considered the core of these beliefs by their practitioners.
Around the same time that the mainland and Eleatic Greeks were developing philosophy - let's just say ca. 450 BCE - Indians were independently and in parallel elaborating many similar ideas in nastika belief. It is hard to know how far back these traditions go, or in what form they existed, for various reasons, but the best-known teachers are thought to have lived around the same time as the best-known early Greek teachers. There are four prominent beliefs, two of which still clearly exist in the widely observable modern world. Each had its own view of "enlightenment" and the "karmic" cycle (samsara). I will make some rather contentious comparisons to Greek philosophy.
Buddhism
Buddhism in its modern widespread form often involves idol worship, but Siddhartha (the Buddha) is supposed to have said to know the way was to know him (whereas Jesus is said to have said 'the way is through me', could this mean the same or the opposite?). I wonder what the Buddha would think of having this belief system named to his memory. Perhaps it wouldn't matter. Buddhists often subscribe to four noble truths, and they often expect to escape from the six realms of existence after spinning around through them, with escape being its own sort of boundless 'realm'.
Anyway, the Greek counterpart I think of is Socrates, who did not expect an unsuffering life, and who seemed to perceive his end as an escape into another unbounded realm.
Jainism
Jainism's best-known teacher is Mahavira. Jains typically subscribe to the seven tattva (I think the number varies). A principal of non-violence is strong in Jainism. The Jainist stories are sometimes seen to involve magic substances, karmic 'leshya'. Jainism claims to be the oldest of the nastika traditions, I believe, although I don't know if this is verifiable. Pali teachings recorded by the Jains have preserved a great deal of ancient history, although this history bears the mark of some alteration. Jainist ideas of enlightenment, as I understand them, involve the idea of sort of leveling up to connect with all spacetime as a sort of circuit; in this way, it is a faith of individual empowerment and freedom, as well as human connection. Jainism strives for actions that are Punya (virtuous, free from expectation), as opposed to Papa.
The Western philosopher I associate with Jainism is Anaxagoras, who had a sort of magical and all-encompassing understanding of the universe.
Ajivika
The Ajivika belief is probably the least known, but it comes next, because it has the closest ties to Jainism, and historically it was pronounced. The Jains recorded some of their ideas, mostly to show them wrong. A prominent Ajivika teacher Gosala was said to have walked and argued with Mahavira, until he finally killed himself after losing an argument, although it seems I've heard accounts where he is said to have drowned himself, and others where he starved. (He was so much of a wimp that, as he starved to death, a follower offered him a mango pit, and he sucked on it! (sarcasm)) The Ajivika faith was said to have been wiped out in a single purge, in which I think something like 18,000 believed followers were killed, and every writing that could be found was destroyed, so the story goes. I believe there are still practicing Ajivika in both Northern and Southern India. They seem to have been considered somewhat annoying in history, even though they were considered somewhat passivist. The story goes that starving people would show up in villages unclothed, and then when offered food, would give much of it away to animals and other people. They didn't seem to look down on dealings with prostitutes. In addition, they were said to practice removing their finger joints for some kind of life-force ritual, as well as other strange rites. For them, the understanding of the cycle of time can be compared to unwinding string (karma, or at least the idea of karma) from a yo-yo. Eventually the string just falls away after a certain time. The Ajivika were determinists; it didn't matter too much what one did. They were also amazingly specific in terms of quantification, and were atomists. For instance, of the jiva (life) atom that was said to leave the body at death: it could not be split, was of eight parts, the color of palai fruit, and extended 2500 miles. This sounds like a completely self-contradictory statement - how could it be true?! But we now accept that there are atomic subshells that are indestructibly one piece but 8 sections and extend indefinitely theoretically. Another sort of strange embodiment for an atheistic faith, according to the existing writings: when a person is finally reaching the end of the cycle to become nothing, they will reach a doorway; Manibhadra and Purnibhadra will swoop down and try to hold on, but the person who releases will not get wound up again in the karmic cycle.
The Greek analogue I think of is Leucippus. Although we don't know much about him, we don't know much about the Ajivika either. He was obviously an atomist, and he seemed a bit of a humble wandering spirit, a strange mix of realism and fantasy.
Lokayata
'Worldly ones', also known as Charvaka ('sweet-talkers') or Brihispatiya. According to Vedic tradition, Brihispati originally taught this philosophy. It is not a good thing to be called, though, because he apparently did this to trick the asuras, power-seeking deities, so they could be more easily conquered. There's another story that Brihispati broke the head of Gayatri; each fragment became a new head, and the brain matter that fell to earth became differently colored cows, so that perhaps she represents the elements. Anyway, the Lokayata believed in the traditional elements, which were more or less the four Greek elements. Some Lokayata may have argued that death just meant a splitting of elements. Basically they were skeptical naturalists, and they are recorded as having been a pretty contentious philosophical force. They believed that the elements could combine in innumerable ways. I believe the Natya scales and elements of Indian theatre bear their mark. Their best-known teacher in ancient times was Ajita, I believe.
The Greek I would compare is Empedocles, who developed Greek elementalism, and who wrote in verse, a sort of endless speciation of ideas.
