Sutra and tantra run together, from the Madhyamaka view all the way up to highest yoga tantra.
Tibetan Buddhism is the largest system of the three transmissions: it preserves both the scholastic tradition of late Indian Buddhism and its tantric transmission, in full. A Geluk monk spends twenty years on logic, Madhyamaka, prajñāpāramitā, Abhidharma and vinaya before entering tantra at all.
This page does no more than the basic sorting: what is most recognisable about each of the four schools, plus the Jonang, which usually gets left off the list. Every Tibetan school could fill an entire site.
Sutra and tantra together: the correct Madhyamaka view as the ground, highest yoga tantra as the path. Sutra establishes the view, tantra supplies the fast method, and neither works without the other. On this the Tibetan schools agree.
The stages of the three capacities: the lesser scope (aiming at a good rebirth), the middling scope (aiming at individual liberation) and the greater scope (aiming at complete buddhahood). Tsongkhapa's Lamrim Chenmo lays the whole road out along this framework as a single line.
Guru yoga as the hinge: Tibetan Buddhism weighs the teacher-student relationship far more heavily than East Asian Buddhism does. That is why its transmission holds so well, and it is also where things go wrong most easily.
By Tsongkhapa. Widely available in translation and the standard way in for readers coming from outside. It arranges the whole road, from the first resolve to buddhahood, into a sequence you can act on.
Candrakīrti and Nāgārjuna. The shared basis of view across all the Tibetan schools.
Brought together by Longchenpa (1308–1364). The terma (treasure) tradition is a way of sourcing texts particular to the Nyingma.
The realisation songs of Milarepa (1040–1123). The most affecting work in Tibetan literature, worth reading whether or not you practise.
Descended from Virūpa in India, the core teaching of the Sakya school.
By Sogyal Rinpoche. Strictly speaking not a traditional treatise, and the book through which more people worldwide first met Tibetan Buddhism than any other. It covers the bardo and care for the dying.
Formed in the 11th century, with the teaching traced back to Padmasambhava in the 8th. Key figures: Padmasambhava, Longchenpa, Jigme Lingpa. The core is trekchö (cutting through) and tögal (direct crossing), on the premise that intrinsic awareness is already complete and needs no contrivance. The nine-vehicle classification and the terma tradition are its distinctive features.
11th century. Lineage: Tilopa → Nāropa → Marpa → Milarepa → Gampopa. The six yogas of Nāropa (inner heat, illusory body, dream, clear light, bardo, phowa) are the most recognisable combination of tantric methods. The tulku system was invented by the Karma Kagyu. Karma Pakshi (1204–1283) was identified as his predecessor's rebirth during his own lifetime, and the institution was fixed with the recognition of the third Karmapa, Rangjung Dorje, born 1284.
Sakya monastery founded in 1073. The five Sakya patriarchs, of whom Sakya Pandita and Phagpa were politically central under the Yuan. The core view is the inseparability of saṃsāra and nirvāṇa, and the system of practice is Lamdre.
Tsongkhapa founded Ganden monastery in 1409. The heaviest emphasis on sequence, discipline and monastic education: a geshe degree takes more than ten years of debate. Prāsaṅgika Madhyamaka is held as the only definitive view. The Dalai Lama and Panchen Lama lines belong to this school.
The fifth school, usually left off the list, and still in existence today (Rangtang in Sichuan, parts of Qinghai). Known for the view of extrinsic emptiness and for the six-branch yoga of Kālacakra. It was pushed to the margins for a long period after the rise of the Geluk.
First: the tulku system is not a general Buddhist institution. It is an institutional invention of 13th-century Tibet, originating with the Karma Kagyu. Neither East Asian nor Theravāda Buddhism has anything like it.
Second: the word "Lamaism". It is a name applied from outside, by the Qing and by Westerners, never used of themselves, and it carries the implication of placing the tradition outside Buddhism.
Third: Tibetan tantra and Tangmi are distinct. Tibetan Buddhism includes highest yoga tantra; Tangmi and Japanese Mikkyō belong to the middle period of esoteric Buddhism and have no such class. That is the key to the difference in content.
One practical warning that has to be said plainly: Tibetan Buddhism asks for a very high degree of trust in the teacher. That is highly effective for transmission, and it also makes it the system most open to abuse. Recent decades have produced a number of international cases of sexual and financial misconduct involving well-known teachers, and the Dalai Lama himself has said publicly, more than once, that students must examine their teachers and expose them where necessary. "Whatever the guru says is right" is not Tibetan doctrine. "Observe for a long time before you commit" is.
The standard way in is the Lamrim Chenmo, with a great deal of commentary available around it. It requires no tantric background and its structure is clear.
If you want to get the feel of it first, read The Hundred Thousand Songs of Milarepa. A man who bet everything he owned wrote it, and it reads nothing like a religious document.