Teachings from Buddhas other than Gautama

The Buddhavamsa (a very late addition to the Pali canon) gives accounts of the 24 previous Buddhas. For each of the past 24 Buddhas, it describes the "three occasions of the Buddha's teaching".

They all started by teaching the same sutta (Dhammacakkappavattana Sutta) this is always the first occasion of a Buddha's teaching.

The second and third occasions of a Buddha's teachings vary according to the Buddha involved. For nine of the 24 previous Buddhas, the second occasion was going to Tāvatiṃsa heaven to teach the Abhidhamma.

Some of the previous Buddhas also taught suttas such as the Mangala Sutta.

There is no evidence external to the Buddhists texts that Gautama ever existed, late references to the Buddha as Viṣṇu in some Purāṇas are part of a move to assimilate the Buddha into Vaiṣṇavism (not before the 8th Century CE). My unpublished article on The Buddha's supposed names, for example, shows that these names were made up some time later for an audience familiar with and probably immersed in Brahmanical culture norms: Siddhārtha Gautama: What's in a Name?

For the lineage of past Buddhas there is no evidence at all, except for some texts composed by Buddhists to support the assertion. The only people who take these texts as anything like history are religious Buddhists. They have precisely the same status as other religious texts which claim to have knowledge beyond the human sphere. We have to take these texts alongside, say, the Genesis chapter in the Bible as myths specific to a time and place. To take them literally is naive at best.

The consensus of scholars is that the stories about the past Buddhas are part of a myth which seeks to make Buddhism seem more authentic to the Iron Age Indian audience by making it seem to be a lineage of teachings. This story exists in tension with the idea that Buddhism is an entirely new teaching. A key article in this line of thinking is Richard Gombrich's 1980 article "The Significance of Former Buddhas in the Theravadin Tradition" (online courtesy of the Oxford Centre for Buddhist Studies).

"[The Buddha] differed from [Brhamins] in his failure to authenticare his position by reference to a long line of teachers. Sages proclaiming bold ideas in the Upaniṣads at least presented their credentials by listing their teachers, their geneaology in sacred lore, all the way back to omse mythical culture hero such as an inpsired reciter of the original Veda. The fabrication of such a genealogy for the Buddha, however, present insuperable problems... [because he had no teachers]."

"The Buddha's authentication could therefore only be mythological. It is here that we find the fundamental raison d'etre for the doctrine of previous Buddhas. A model lay close at hand, in Jainism... that the analogous Buddhist doctrine of previous Buddhas was influenced by Jainism seems much more likely than that the Jains borrowed their doctrine from the Buddhists." (Gombrich p.64)

Another article to read, if you can get hold of it is, Naomi Appleton's The Multi-life Stories of Gautama Buddha and Vardhamana Mahavira. Buddhist Studies Review. Vol 29, No 1 (2012) [Link is to abstract]

Note also that the lineage lists found in the Bṛhadāranyaka Upaniṣad list many men named Guatama. It was a high status Brahmin name dating back to the Ṛgvedic period (ca 1500-1200 BCE). It's highly likely that the Chan/Zen lineages that link Chan patriarchs back to the Buddha are also fabrications for the same reasons. On this subject see John McRae's book Seeing Through Zen: Encounter, Transformation, and Genealogy in Chinese Chan Buddhism.

Given this, We would not expect such mythic figures to have passed down actual teachings. The teachings we have probably date from a period some centuries after the putative life of the Buddha. However as the other answers to this question show, stories were fabricated and authenticated to make this lineage seem plausible. Gombrich discusses most of these texts in his article.

The Mahāyāna Sukhāvativyūha Sūtras purport to be teaching by or about Amitābha Tathāgata, who lives in another universe than out one, but is able to intersect with our universe at the point of death and guide a being to be reborn in his universe where enlightenment is very easy. A number of other Pure Land sūtras explore similar themes.

In Theravada tradition, a Buddha appears in this world only after the teachings of the previous Buddha completely vanishes. It is also mentioned that the teachings of a Buddha will survive after his parinibbāna only if he teaches the Vinaya pitaka.

Yes, the historical Buddha is said to have related teachings of other Buddhas; one that stands out is the Amagandha Sutta, which relates the teaching of Kassapa Buddha. Another is MN 81, which relates some aspects of Kassapa Buddha's life, and some words of Kassapa Buddha:

“The king thought: ‘The Blessed One Kassapa, [51] accomplished and fully enlightened, does not accept from me a residence for the Rains in Benares,’ and he was very disappointed and sad.

“Then he said: ‘Venerable sir, have you a better supporter than I am?’—‘I have, great king. There is a market town called Vebhalinga where a potter named Ghaṭīkāra lives. He is my supporter, my chief supporter. Now you, great king, thought: “The Blessed One Kassapa, accomplished and fully enlightened, does not accept from me a residence for the Rains in Benares,” and you were very disappointed and sad; but the potter Ghaṭīkāra is not and will not be so. The potter Ghaṭīkāra has gone for refuge to the Buddha, the Dhamma, and the Sangha. He abstains from killing living beings, from taking what is not given, from misconduct in sensual pleasures, from false speech, and from wine, liquor, and intoxicants, which are the basis of negligence. He has unwavering confidence in the Buddha, the Dhamma, and the Sangha, and he possesses the virtues loved by noble ones. He is free from doubt about suffering, about the origin of suffering, about the cessation of suffering, and about the way leading to the cessation of suffering. He eats only one meal a day, he observes celibacy, he is virtuous, of good character. He has laid aside gems and gold, he has given up gold and silver. He does not dig the ground for clay with a pick or with his own hands; what has broken off riverbanks or is thrown up by rats, he brings home in a carrier; when he has made a pot he says: “Let anyone who likes set down some selected rice or selected beans or selected lentils, and let him take away whatever he likes. He supports his blind and aged parents. Having destroyed the five lower fetters, he is one who will reappear spontaneously [in the Pure Abodes] and there attain final Nibbāna without ever returning from that world.

(from Middle Length Discourses of the Buddha, Bodhi, trans)

There are probably others I can't think of off-hand, but they all would have to have come from the current Buddha, since the interval between Buddhas is invariably so vast as to not allow for any trace of the previous Buddha's teachings to remain.

(This all from a Theravada Buddhist POV)