The distinguishing teaching about the Upanishads is that 1. God is not a person 2. The universe, as we perceive, is an illusion and its underlying reality is God 3. The individual (a person like you and me), is not just a collection of bones, muscles and tissue. Neither is the individual his/her mind. The true-nature of an individual is the indestructible Self. 4. The individual Self and God are one and the same.
There's no such teaching like this in the world. It blows my mind as I read and contemplate on the verses that talk about it. There are several saints who have realized this truth and their position on the issue of God, individual and world are identical to what the Upanishads teach.
IMHO A person who has put to test the teachings of the Upanishadic in his/her own life, and lives by its vision is a real teacher.
The Mandukya Upanishad talks about "ajata-vada", ie the argument that nothing is ever born. Its teaching is very close to buddhist philosophy.
The Mundaka Upanishad talks about the creation being God itself. A famous mantra quotes an example of a spider which is not only the efficient but also the material cause of the webs that it spins.
Chandogya Upanishad chapter 6 has the "maha-vakya" i.e. great statement "tat-tvam-asi" - thou art that.
Of course, reading a translation will not be sufficient to understand the depth of this text. These texts are usually expounded by scholars who are steeped in the tradition of studying and understanding them. For advaita (non-dualistic) philosphy, the master-commentator Shankaracharya has written brief commentaries on all the Upanishads. Their translations are available in English, but like I said earlier, a mere reading of them may not bring about the desired affect of "oh my god this is so awesome".
http://www.uesp.net/wiki/Lore:CHIM
http://www.uesp.net/wiki/Lore:The_Monomyth
http://www.uesp.net/wiki/Lore:Spirit_of_Nirn,_God_of_Mortals
It's ultimately not all that deep, but I'm fairly sure it's the only top-selling mass-market video game series to touch on the concept in even that much depth.
http://www.jkrishnamurti.org/about-krishnamurti/dissolution-...
"I maintain that Truth is a pathless land, and you cannot approach it by any path whatsoever, by any religion, by any sect. That is my point of view, and I adhere to that absolutely and unconditionally. Truth, being limitless, unconditioned, unapproachable by any path whatsoever, cannot be organized; nor should any organization be formed to lead or to coerce people along any particular path. If you first understand that, then you will see how impossible it is to organize a belief."
"Again, you have the idea that only certain people hold the key to the Kingdom of Happiness. No one holds it. No one has the authority to hold that key. That key is your own self, and in the development and the purification and in the incorruptibility of that self alone is the Kingdom of Eternity."
Note the context -- he was himself "chosen" to be "the guru" as he was 16(!) and he recognized the absurdity of it as he was 32:
"The Order of the Star in the East was founded in 1911 to proclaim the coming of the World Teacher. Krishnamurti was made Head of the Order. On August 3, 1929, the opening day of the annual Star Camp at Ommen, Holland, Krishnamurti dissolved the Order before 3000 members."
( The film of him reading the speech then: https://www.youtube.com/watch?v=VYnxRYAHmEs&t=11s )
The teachings of the Order in which he was raised weren't "Upanishads and Gita" but the "Doctrine" of Madam Blavatsky:
https://en.wikipedia.org/wiki/Order_of_the_Star_in_the_East
"One of the central tenets of late 19th-century Theosophy as promoted by the Theosophical Society was the complex doctrine of intelligent evolution of all existence. This was said to be occurring on a Cosmic scale, incorporating both physical and non-physical aspects of the known and unknown Universe, and affecting all of its constituent parts regardless of apparent size or importance. The theory was originally promulgated in the Secret Doctrine (published 1888), a book by Helena Blavatsky, one of the founders of contemporary Theosophy and the Theosophical Society."
Edit: Upanishads have, unsurprisingly, many more different teachings than what "tech_browser" wrote below:
https://en.wikipedia.org/wiki/Upanishads
They are huge and with a lot of stories, a random sample is here:
http://www.sacred-texts.com/hin/sbe15/sbe15075.htm