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I've been practicing with the Triaratna Community for over 7 years and over that time I've been through a fair number of texts with various study groups. I'll try to summarise the most relevant texts here but different order members/centres certainly have their favourites so someone else in the community might give a slightly different list.
As a bit of background, the Triratna Community is an eclectic school so our texts come from all the branches of Buddhism. However the founder Sangharakshita studied and practiced Buddhism in the 1950s in northern India around the time that there was an exodus of Tibetan lamas from Tibet. So in practice the texts we study tend to be influenced by Tibetan or Theravadan traditions. Zen and Chinese Buddhism in general is under represented. Not that we think there is anything wrong with those texts but the history of the movement means we have less exposure to them.
So the texts we study tend to be either from the Pali Canon or Mahayana texts (transmitted by his contact with Tibetan lamas I think) or texts that Sangharakshita himself has written. We run a four year study course (which I've done part of) which covers the main principles and texts so a browse through that would give you a flavour but I will attempt to summarise the main ones
Pali Canon
Middle Length Discourses generally but more specifically from the pali canon
Mahayana
Also
Also there are a great many books by Sangharakshita which are studied and referenced. If someone was interested I would probably recommend Who is the Buddha, What is the Dharma and What is the Sangha as a good entry point into his thinking. When people are becoming ordained the read and study his book The Survey of Buddhism which is where he first put his ideas down. It's a dense read though so I wouldn't necessarily recommend it unless you were very interested.
EDIT
I've just come across this very comprehensive book, Puja Readings and Other Texts as used in the Triratna Buddhist Community, which details all the texts used by the Triratna movement. As well as been a good reference to the Triratna curriculum, I'd recommend it as a good general anthology of Buddhist texts covering Theravadan, Mahayana and Tibetan sources. Certainly worth a look if anyone is interested and reasonably priced I think. It weighs in a mighty 700 pages plus so there is a lot of reading for your money.
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There is much material available that describes these practices such as https://thebuddhistcentre.com/system/files/groups/files/Triratna%20Resource%20Pack%20for%20Newcomers%20-%20texts.pdf
Here is a brief summary https://thebuddhistcentre.com/text/what-triratna-buddhist-community
The Triratna Buddhist Order and Community is a worldwide movement of women and men living by the Buddha’s teachings in the conditions of the modern world. Neither monastic nor lay, we are simply Buddhists, at varying stages of commitment and understanding.
Triratna is a Sanskrit term meaning ‘Three Jewels’: the Buddha, Dharma (his teachings) and Sangha (the community of all those who follow the teachings). The founder of the Triratna Buddhist Community and Order, Sangharakshita, considers the defining act of a Buddhist to be Going for Refuge to these Three Jewels. This is the central principle or orientation of the Triratna Buddhist Community and everything we do. At our Buddhist Centres we teach meditation, study the Buddha’s teaching together, engage with the Arts, support each other through life, and engage in our local communities. We also promote projects in which Buddhists can live and work together, and explore how to turn our work into a spiritual practice.
If you are searching for a center near you https://thebuddhistcentre.com/text/triratna-around-world