I don't think you should discount poetry, charms & songs so quickly. In today's world, and especially in anglo-american philosophy we expect philosophy to be in prose & discursive.
This wasn't true when philosophy first began. It arose out of mythology, and the two worlds intertwined for some time. As an example, Empedocles & Parmenides wrote their philosophy in verse (as did mathematicians).
This tradition still continues in continental philosophy, where poetic values are still important. Nietszche wouldn't have written the way he did if not for that.
A general problem for the Western understanding of Indian philosophy is that it's not taken as philosophy. It's studied either in philology, theology & Sanskrit studies. To be fair philosophy is not a discipline apart - it interacts with other expressions of culture.
The Upanishads are taken to be the beginning of Indian Philosophy, astika & nastika split philosophy into two broad camps, those that were inspired by the Vedas, and those that weren't. It's not synonymous with theist/atheist in western terms as you can get atheistic philosophy that conforms with the Vedas such as Samkhya.
Atomism rose as early on as a philosophical position, and were investigated by the Ajivika, Carvaka, Nyaya & Vaisheskika schools within the context of Jain, Buddhist & Veda religious traditions. The trigonometric functions such as sine were taken from the 3BCE Surya-siddhanta a work of astronomy. The 14C Kerala school of mathematics investigated the notion of a power series well before the invention of calculus in Europe, but not before the 10C Iraqi mathematician al-Hazen.
The Catuskoti is important in Classical Indian Logic. It states for a proposition p:
- not p
- p and not p
- neither p nor not p
Within the Western Philosophy the importance of the first two is undisputed. The other two have been very strongly disavowed, but recently they've formed their own schools as in dialethism and paraconsistent logic.
A good book, I've been told, is the scholarly work by McEvilley titled Shape of ancient thought which discusses both the ancient Greek & Indian philosophy and how they've mutually affected each other. Another is Westerhoff's Nagarjuna's Madhyamaka.