This is a great question, and one with a relatively high barrier to entry if you don't know where to start. Even a lot of secondary sources just cite secondary commentaries that are often from scholars working off original language texts, which can make finding primary translations more difficult.

Avesta.org is going to be your new best friend. It's a trip back the internet of 1998, but a resource compiled for Zoroastrian believers with the full (surviving) Avesta and many of the other key works of Zoroastrian theology. Some of them, including the same translations of the Avesta, are also available from sacred-texts.com, which has a more readable interface imo. If you absolutely want the full Avesta in print this 3 volume printing is the only one I know of with a reliable translation (which is the same as those online sources I linked to). Sacred-texts, the print edition, and most of the translations on Avesta.org are all public domain for two reasons. 1: It's public domain and free to post online. 2: The 1880-1887 translation project by James Darmetester and L.H. Mills is the only translation of the full Avesta in English. Individual sections have more recent translations, but those 19th Century ones are the only full version.

What is the Avesta?

The Avesta is the most ancient collection of Zoroastrian hymns, prayers, laws, and liturgy, and its antiquity gives it a status similar to Biblical canon in Zoroastrianism. Later religious works are still valued, studied, and respected but not to the same degree. The etymology of the name isn't clear, but it might derive from a phrase that meant "works of praise."

The "real" Avesta should not be confused with the Khordeh Avesta (literally "little Avesta"), which is actually more like a genre itself. Different priests and religious scholars produce Khordeh editions with different texts included, some from the Avesta but also some from later tradition. The Khordeh functions as a sort of book of common prayer, with most of the daily and holiday prayers, but also as a devotional for personal study, which is why different writers include different selections.

What does it consist of

The Avesta contains all of the surviving writing in the Eastern Iranian language called Avestan, or more accurately two languages or stages of linguistic development. The oldest sections are written in Old Avestan, which linguists estimate was spoken around 1400-1000 BCE. The bulk of the text is written in Younger Avestan, which evolved out of Old Avestan and was spoken from about 900-400 BCE. Historically, Avestan was also called "Zend," which was actually the Zoroastrian name for commentaries written about the Avesta. That's why you'll sometimes see older scholarship, like the 19th Century translations, call it "The Zend Avesta."

The modern Avesta is the end product of centuries of development from various separate works created over about 800 years and only compiled together as a single canon much later. Zoroastrian tradition and modern scholarship alike agree that it was passed on through oral tradition for most of its history. Tradition and academia diverge on when exactly the Avesta was codified as a single written corpus.

how is it organised?

The core of the Avesta is the "Yasna," which is recited in its entirety at formal Zoroastrian worship ceremonies. The Yasna itself is basically a grab bag of different styles and genres of religious literature ranging including prayers, invocations of divinities, and statements of belief called matras (cognate to the Hindu concept of a mantra). Most sections of the Yasna have their own titles, which usually indicate what type of prayer or ritual they are.

The climax of the Yasna ceremony is centered around the "Gathas," the oldest section of the whole Avesta. Many parts of the Avesta are framed as a dialog between the prophet Zarathustra and a divinity, but the "Gathas" are the section that is almost always understood to be the actual composition of Zarathustra himself. In fact, even modern historical and linguistic analysis agrees that they are both the oldest Avestan hymns and that they are stylistically consistent enough to be the work of a single author or a small group with intentional conformity. They don't identify their own composer, but they are so old and consistently attributed to Zarathustra that it's reasonable to identify him as the author.

The Yasna is supplemented by the "Visperad." Like the Yasna, it isn't a single work, but a series of prayers, hymns, and commentaries meant to reinforce parts of the Yasna by interweaving each chapter of the Visperad into the Yasna recitation on the six annual gahambar festivals.

The "Vendidad" is the third component that can sometimes be added into the Yasna ceremony on certain festivals, usually Maidyarem (mid-winter) and Hamaspathmaidyem (the days immediately before Nowruz, the Zoroastrian new year in Spring). Unlike the first two sections, the Vendidad is not liturgical. Instead, it is a law code for ritual purity with a couple of mythological stories at the start.

The last major section of the Avesta is the "Yashts," which are prayers and invocations dedicated to Zoroastrianism's many divinities including Ahura Mazda, the six Amesha Spentas together, some of the Amesha Spentas alone, the guardian spirits called fravashi, and 15 to various yazatas. They cover offerings of sacrifice, ceremonial worship, mythology, and requests for protection. Unlike the other three large sections, the Yashts are not part of the standard worship ceremony, but get used in personal devotions, holiday festivities, and other specialized ceremonies.

The remaining sections of the Avesta are largely related to day to day prayer. The "Gahs" are the five daily prayers. The "Afrinagans" are payers for the six Gahanbar festivals. The Zoroastrian calendar associates each day of the month with a different divinity, and the "Sirozahs" are the prayers to each divinity recited on their day. The "Nyayis" are associated with different times of day or astrological events and recited at the event they're focused on, either daily or monthly. Most of these are partially composed of quotes from the larger sections of the Avesta.

What other religious works other than the Avesta exist?

There are many. Zoroastrian religious writing didn't really take off until around 300 CE, but that still gives us 1700 years of writing to include in this category. None of it holds the same place in theology as the Avesta itself, but many have significant spiritual importance, especially those that include discussion of books from the Sassanid Avesta that have been lost. They are written in many languages.

The Denkard and Bundahishn are probably the most significant, with the former providing summaries of the Avesta (including lost books) and the writings of pre-9th Century priests and the latter describing cosmology, natural philosophy, and eschatology. The Bundahishn was preserved in India, while a slightly different and much longer version called the Greater Bundahishn was preserved in Iran and includes writing added on as late as the 13th Century. The Dadestan-i Denig is the other, long, significant medieval work, providing a priests commentary on common religious questions, framed as dialogs between himself and various questioners.

The "Rivayats" and "Zand" are whole genres of commentary on other religious writings, composed in all of the relevant languages. Zand typically indicates commentary on the Avesta itself, while the Rivayats are usually longer and provide commentary on non-Avestan texts. Some are more prominent than others, and those are usually the ones available in translation.

You can follow Zoroastrian literature all the way up to the present day, with priests of all ranks, but usually Dasturs, continuing to write new commentaries, devotionals, and other works. The most significant example since colonial European contact is probably Dastur Meherjirana's Rehbar-e Din-e Jarthushti, a late 19th century handbook that laid out Meherjirana's influential reformist theology.