Okay, onto the complications! Let's begin with the idea of the Islamic Golden Age, and, relatedly, the idea of "what went wrong". I'm not positive I know exactly what you're referring to when you say "what went wrong", but I suspect that you may be suggesting that after the Islamic Golden Age and its associated scientific learning, there was a period of decline vis-a-vis said learning in Islamic lands. I expect many people would say "the siege of Baghdad by the Mongols in 1258 is what went wrong."

But, you have me here, and I want to start out, first and foremost, by complicating both the concept of a 'golden age' and the notion of decline. Declines and golden ages go together--for there to be decline, there must be progress, advance, and innovation first; for there to be a Golden Age, there must be other, less-golden ages bracketing it. So, what's wrong with that?

There's three things I want to point out here. First, the idea of 'decline' historiographically is deeply value laden, emerging originally from 16th Italian diplomats who held cyclical understandings of history and used those understandings to argue that the Ottoman Empire would soon fall--it isn't a given that civilizations or cultures have 'golden ages' or 'declines'; other historians have made those categories and what they assign value to as 'golden' or as 'decline' tells us usually less about the culture being studied and more about the people making those definitional categories. Second, these big historical and era-defining categories presuppose that vast regions populated by a lot of different peoples across hundred of years all understood the same developments/social changes/values/etc as cohesive, as meaning the same thing. It suggests that all those people from the 8th to 13th centuries in what was the biggest city in the world at the time all shared the same standards, norms, and approaches to scientific thinking and learning. Dear reader, they definitely didn't.

Finally, third, and this is my last point on declines and golden ages, and also maybe the most important one: notions of decline and golden ages assume a universal history of science wherein all previous and separate past cultures work towards 'modern' science. I'm going to quote Sonja Brentjes here, from her article "The Prison of Categories--Decline and Its Company" (from which I first learned most of these ideas):

"We need to go further and abandon...a universal history of science in which all the separate cultures of past science contribute towards 'modern' science, each in its own way. There is only one way in which a universal history of science can be maintained--by the recognition that all science is culturally constructed. As cultural constructs, the sciences of different societies can be measured and compared in terms of their complexity, degree of difficulty, and explanatory and prognostic capability, but also their positive as well as negative impact on society, nature, and the universe. But they are not Aristotelian forms which reach their full state in our own sciences. They do not grow in a cumulative manner towards us. Our knowledge and our practices do not comprise of all earlier stages of scientific doctrines and activities (pages 137-138)."

So the reason I'm giving you all this information which may seem somewhat irrelevant to "who were the key figures who advanced scientific innovation in the ʿAbbasid Golden Age?" is because these ideas of golden age and decline in fact deeply underscore that question and what its answer means to us. We are conditioned to think that science was one thing, that its development was linear, and that innovation is its pinnacle value. Part of my job as a historian of science is to chip away at that conditioning. I hope as you pursue this topic/the history of science more, you'll keep reflecting on these ideas!

Tied to all of this is also the notion of 'advancing' that you bring up, OP: advancing human understanding and thinking. I like the phrasing of "advancing human understanding" because it has the potential to be wonderfully capacious, and I think scientists/innovators/thinkers of the past all worked towards a development of knowledge with the same rigor we do today; we just sometimes rely on different fundamental understandings, principles, or methods in the process. I flag this, though, because of what I said (/what Brentjes said) before about a universal history of science: while contemporary science has built upon foundations laid by these thinkers, it's not a straight linear shot to today's understanding of the world, and they didn't do all that thinking just to get us where we are today. So, in all truth, I can't tell you how far these thinkers advanced because I can't possibly bound that fairly; I wouldn't know on what scale to judge, nor do I want to make that sort of judgement.

I hope all this helps answer your question, OP. I also hope it makes you even more curious about this topic and that this proves to be the start of a longer journey of academic exploration for you. The history of Islamic science, philosophy, and thought is rich, complex, and compelling. Happy reading!

Hey OP! I'm going to try to answer your question not only by providing you some of the specific data you're asking for but also by complicating a few pieces of your question itself. I hope this will give you solid foundations to build upon as you continue to explore the history of Islamic science and thought!

Just to lay out a few basics for everyone: generally speaking, the Islamic Golden Age refers to a period of flourishing in the history of Islam from the 8th to 13th centuries. The ʿAbbasid caliphate extended, at its largest point, from modern day Tunisia to modern day Afghanistan. It's worth noting that when the Golden Age is discussed, though, the conversation usually centers around Damascus to Isfahan, with Baghdad in the middle. The beginning of said Golden Age is marked by the ʿAbbasid caliph Harun al-Rashid and ends with the Mongol invasion of Baghdad in 1258.

Let's talk about the data at the center of your question before I take this opportunity to talk your ears off about methodologies of the history of science (but promise you'll read all the way to the end--it's important!). Limiting ourselves to that 8th-13th century ʿAbbasid period--which is a temporal and geographic boundary!--who were the big names and what did they do? Well, okay, I actually need to fuss a little--'scientists, innovators, and scholars' is a HUGE collection of people and an even larger collection of disciplines! It's also worth noting that what was considered 'science' now didn't necessarily refer to the same things that were categorized as science in that time frame (i.e. grammar and theology were both sciences; heck, there's even a whole debate about artificial poetic figures that uses the terminology of innovation within the science of prosody). And the "big names" sometimes don't actually keep to those temporal and geographic boundaries (or they keep one but not the other). This is all to say--a single Reddit post probably can't answer your question. To make a start, though, I'm going to list some figures and texts from the disciplines that most closely relate to contemporary STEM fields and philosophy. I'm vaguely and gently keeping the geographic and temporal boundary we've established but I'm prioritizing the "bigness of name" over upholding that. I'm also going to give you more sources than anything else, because I don't actually have the capacity to cover all these categories in summary. I have chosen sources that are fairly accessible and that will give you good information on a topic-by-topic basis. NB: this is *not* an exhaustive list!

STEM:

For astrology and astronomy, start here: http://www.metmuseum.org/toah/hd/astr/hd_astr.htm . Some names include: Abu Maʿshar al-Balkhi, al-Biruni, Nasir al-Din al-Tusi, ʿAbd al-Rahman al-Sufi, and Ibn Haytham. A good starter secondary source is A history of Arabic astronomy by George Saliba.

For engineering, machines, and technology: Ismāʿīl ibn al-Razzāz al-Jazarī is a big name; check out Islamic Science and Engineering and Studies in medieval Islamic technology by Donald R Hill. You can also read Islamic technology: an illustrated history by Ahmad Yusuf Hasan.

For alchemy and chemistry, see my other AskHistorians post!

For botany and agriculture, see Abū Ḥanīfah Dīnawarī's Book of Plants (trans. Catherine Breslin). Also, Agricultural Innovation in the Islamic world: the diffusion of crops and farming techniques 700-1100 by Andrew M Watson.
These are not the only disciplines in which intellectual and/or scientific contributions were made--I've neglected medicine, zoology, and mathematics, to name just a few.

Philosophy:

Here is the list of the classically biggest names in Islamic philosophy from approximately this period: al-Kindi, Isaac Israeli, Abu Bakr al-Razi, al-Sijistani, al-Kirmani, al-ʿAmiri, al-Farabi, ibn Sina, al-Ghazzali, ibn Bajja, ibn Tufayl, and Ibn Rushd.

If I had to recommend one starting book about Islamic philosophy, I'd choose Philosophy in the Islamic World by Peter Adamson. I will post you a fuller list of philosophy texts in another comment.

A fuller philosophy list:

Altmann, A., and S. M. Stern, Isaac Israeli: A Neoplatonic Philosopher of the Early Tenth Century

Arberry, A. J., Aspects of Islamic Civilization, pp. 119-154 (selections by Abu Bakr al-Razi, “Apologia Pro Vita Sua”, al-Farabi, “The Canons of Poetry”, Ibn Sina, “Life of a Philosopher.”)​

Adamson, Peter, The Arabic Plotinus: a Philosophical Study of the 'Theology of Aristotle

Adamson, Peter, and Richard C. Taylor, eds, The Cambridge Companion to Arabic Philosophy​

Averroes on Plato’s Republic, trans. R. Lerner.

Butterworth, Charles E., ed., The Political Aspects of Islamic Philosophy

​Galston, Miriam, Politics and Excellence: The Political Philosophy of Alfarabi.

Gutas, Dimitri, Greek Thought, Arabic Culture

Ibid, “The Study of Arabic Philosophy in the Twentieth Century: An Essay on the Historiography of Arabic Philosophy,” British Journal of Middle Eastern Studies29 (2002): 5-25.

Khalidi, Muhammad Ali, ed. Medieval Islamic Philosophical Writings

Al-Kindi, “On Rays,” in The Philosophical Works of al-Kindi, trans. P. Adamson and P. Porman (Oxford University Press, 2012)

Lerner, Ralph, and Muhsin Mahdi, eds, Medieval Political Philosophy, Part One, pp. 22-190 (selections by al-Farabi, Ibn Sina [Avicenna], Ibn Bajja [Avempace], Ibn Tufayl, and Ibn Rushd [Averroes]).

Mahdi, Muhsin, Alfarabi's philosophy of Plato and Aristotle

al-Rāzī, Abū Bakr, Spiritual Physick (al-Ṭibb al-rūḥānī), trans. Arberry

Ibn Rushd, Tahāfut al-tahāfut, trans. Van Den Bergh.

Plotinus (Arabic version), Theologia, trans. G. Lewis.

Peters, F. E., Aristotle and the Arabs.

Rosenthal, Franz, The Classical Heritage in Islam (selections, pp. 1-161)

Rowson, Everett, A Muslim Philosopher on the Soul and Its Fate: al-ʿĀmirī’s al-Amad ʿalā l-abad.

​

And some more historiography recommendations:

“The Appropriate and Subsequent Naturalization of Greek Science in Medieval Islam” by A. I. Sabra

Ahmed Dallal’s book Islam, Science, and the Challenge of History

Kapil Raj’s “Thinking Without the Scientific Revolution: Global Interactions and the Construction of Knowledge”

Chakanetsa Mavhunga’s book What Do Science, Technology, and Innovation Mean from Africa

The introduction by Francesca Leoni and Liana Saif in the volume Islamicate Occult Sciences in Theory and Practice.