Among the many schools and traditions, what core teachings are shared by all Buddhists?
There is an existent reference to the similarities between Theravāda & Mahāyāna which I will cite:
(I think it also applies to Vajrayāna although I may be wrong in this regard.)
- Whatever our sects, denominations or systems, as Buddhists we all accept the Buddha as our Master who gave us the Teaching.
- We all take refuge in the Triple Jewel: the Buddha, our Teacher; the Dhamma, his teaching; and the Sangha, the Community of holy ones. In other words, we take refuge in the Teacher, the Teaching and the Taught.
- Whether Theravada or Mahayana, we do not believe that this world is created and ruled by a god at his will.
- Following the example of the Buddha, our Teacher, who is embodiment of Great Compassion and Great Wisdom, we consider that the purpose of life is to develop compassion for all living beings without discrimination and to work for their good, happiness and peace; and to develop wisdom leading to the realization of Ultimate Truth.
- We accept the Four Noble Truths taught by the Buddha, namely, Dukkha, the fact that our existence in this world is in predicament, is impermanent, imperfect, unsatisfactory, full of conflict; Samudaya, the fact that this state of affairs is due to our egoistic selfishness based on the false idea of self; Nirodha, the fact that there is definitely the possibility of deliverance, liberation, freedom from this predicament by the total eradication of the egoistic selfishness; and Magga, the fact that this liberation can be achieved through the Middle Path which is eight- fold, leading to the perfection of ethical conduct (sila), mental discipline (samadhi) and wisdom (pañña).
- We accept the universal law of cause and effect taught in the Paticcasamuppada (Skt. pratityasamutpada; Conditioned Genesis or Dependent Origination), and accordingly we accept that everything is relative, interdependent and interrelated and nothing is absolute, permanent and everlasting in this universe.
- We understand, according to the teaching of the Buddha, that all conditioned things (samkhara) are impermanent (anicca) and imperfect and unsatisfactory (dukkha), and all conditioned and unconditioned things (dhamma) are without self (anatta).
- We accept the Thirty-seven Qualities conducive to Enlightenment (bodhipakkhiyadhamma) as different aspects of the Path taught by the Buddha leading to Enlightenment, namely,
- Four Forms of Presence of Mindfulness (Pali: satipatthana; Skt. smrtyupasthana);
- Four Right Efforts (Pali. sammappadhana; Skt. samyakpradhana);
- Four Bases of Supernatural Powers (Pali. iddhipada; Skt. rddhipada);
- Five Faculties (indriya: Pali. saddha, viriya, sati, samadhi, panna; Skt. sraddha, virya, smrti, samadhi, prajna);
- Five Powers (bala, same five qualities as above);
- Seven Factors of Enlightenment (Pali. bojjhanga; Skt. bobhyanga);
- Eight-fold Noble Path (Pali. ariyamagga; Skt. aryamarga).
- There are three ways of attaining Bodhi or Enlightenment according to the ability and capacity of each individual: namely, as a Sravaka (disciple), as a Pratyekabuddha (Individual Buddha) and as a Samyaksambuddha (Perfectly and Fully Enlightened Buddha). We accept if as the highest, noblest and most heroic to follow the career of a Boddhisattva and to become a Samyksambuddha in order to save others. But these three states are on the same Path, not on different paths. In fact, the Sandhinirmocana-sutra, a well-known important Mahayana sutra, clearly and emphatically says that those who follow the line of Sravaka-yana (Vehicle of Disciples) or the line of Pratyekabuddha-yana (Vehicle of Individual Buddhas) or the line of Tathagatas (Mahayana) attain the supreme Nirvana by the same Path, and that for all of them there is only one Path of Purification (visuddhi-marga) and only one Purification (visuddhi) and no second one, and that they are not different paths and different purifications, and that Sravakayana and Mahayana constitute One Vehicle One Yana (eka-yana) and not distinct and different vehicles or yanas.
- We admit that in different countries there are differences with regard to the ways of life of Buddhist monks, popular Buddhist beliefs and practices, rites and rituals, ceremonies, customs and habits. These external forms and expressions should not be confused with the essential teachings of the Buddha.
-One Vehicle for Peace by Ven. Walpola Sri Rahula, (December 1–7, 1981)
As well as the list which is quoted in Unrul3r's answer there are, also, three other lists included on Wikipedia's Basic points unifying Theravāda and Mahāyāna article.
A less canonical list was written by Christmas Humphreys,
In 1945 he drafted the Twelve Principles of Buddhism for which he obtained the approval of all the Buddhist sects in Japan (including the Shin Sect which was not associated with Olcott's common platform) of the Supreme Patriarch of Thailand and leading Buddhists of Sri Lanka, Myanmar, China and Tibet.
Here for example is a link to the Twelve Principles of Buddhism By Christmas Humphreys.
They include (I summarize, see a reference for the more-complete text):
- Self salvation is for any man the immediate task. (Parable of the Arrow)
- The first fact of existence is the law of change or impermanence. (Therefore clinging to form causes suffering)
- The law of change applies equally to the "soul". (Therefore non-self)
- The universe is the expression of law. (Kamma, purifying inner nature, liberation from rebirth, ultimately every form of life will reach enlightenment)
- Life is one and indivisible, though its ever-changing forms are innumerable and perishable. (Compassion and harmony)
- Life being One, the interests of the part should be those of the whole. (Selfishness is caused by ignorance; the four noble truths)
- The Eightfold Path consists of: etc. (The way is to be practised, not just theory: "Cease to do evil, learn to do good, cleanse your own heart: this is the Teaching of the Buddhas")
- Reality is incomprehensible, and a God with attributes is not the final Reality. But the Buddha, a human being, etc. (The Buddha became enlightened and tought that nirvana can be achieved on earth)
- From potential to actual Enlightenment there lies the Middle Way, etc.
- Buddhism lays great stress on the need of inward concentration etc.
- The Buddha said: "Work out your own salvation with diligence". Buddhism knows no authority for truth save the intuition of the individual, and that is authority for himself alone.
- Buddhism is neither pessimistic or "escapist", nor does it deny the existence of God or soul, though it places its own meaning on these terms.
I can't say that "all schools share these teachings" but apparently, at least, these teachings are compatible with all the schools'.
I'd say nothing, there are so many sects, that you can find any Buddhism that lacks one element or the other.
- Meditation (less common, matra and mudra)
- The Historical Buddha. In some forms of Buddhism, the historical Buddha is not so important.
- Monasticism and renunciation.
- Philosophical bent that cares about topics like identity, change and it's application towards dealing with suffering.
But right off the top of my head, Shin de-emphasizes meditation, SGI deemphasizes monasticism, the Buddha of, say the Lotus Sutra, isn't really the same guy as the Buddha of the early Pali texts, and traditional lay Buddhism doesn't involve much philosophy.
The phrase that comes to mind when I'm reading about yet another sect or school of Buddhism, is "recognizable Buddhism."
The common teachings of all Buddhists are:
• The ordinary lot of human beings is to be suffering from a delusion about the real nature of the universe;
• A method for dispelling this delusion was originated by someone called "The Buddha" who employed it to purpose;
• Buddhism offers to teach the adherent this method so as to cease being subject to the delusion.
Other than these rudiments, all other teachings are variably composed, and supposed literal or metaphorical in their application of instruction. They amount to catechisms of formal doctrine without common confidence or consistency as to their meaning.
There might be many school with many different ideas, some following the Buddha and serving the Dhamma and Vinaya, some using it as base for own ideas. Once the Buddha was asked by his former foster mother, Ven Gotami, in regard of how to recognize whether teachings are Dhamma-Vinaya (the name the Buddha gave his founded religion) or not. He answered:
"Gotamī, the qualities of which you may know, 'These qualities lead to dispassion, not to passion; to being unfettered and not to being fettered; to shedding and not to accumulating; to modesty and not to self-aggrandizement; to contentment and not to discontent; to seclusion and not to entanglement; to aroused energy and not to laziness; to being unburdensome and not to being burdensome': You may definitely hold, 'This is the Dhamma, this is the Vinaya, this is the Teacher's instruction.'"
Possible supportive Introdution to Dhamma-Vinaya.
Maybe worthy to think about is, that certain definitions like such as Buddhism and what might count as such... the standards as well as the orientation, are for the most later development, now also strong dominated by lay people and many various ideas and objectives behind.
The lesser the bound and regard torward the ancestor, elders and leaders, to more it goes astray and it, by time, becomes more and more difficult to find common ground, common objectives and common ways. Lucky if finding a paricular Sangha (monastic disciples) of common offiliation which carries on the heritage. It's worthy to note that the Buddha gave the surviving of the truth and way to it, Dhamma-Vinaya, at least more importance than unity, although he gave very concerning the best tools to handle unity and give into it.
[Note: This is a gift of Dhamma and not meant for commercial purpose or other low wordily gains by means of trade and exchange.]