This is too big a question to be covered in a reddit post, literally entire libraries of books have been written on the subject. It is also a question that covers such a large timespan that it encourages generalisation, generalisation being the bane of accurate historical studies. For example many will tell you it is the Roman military machine that is the cause however this would ignore the 400+ years it took for the city to dominate central Italy. This length of time indicates economic not military dominance.
So your question should more accurately be broken down into sections. I have added a rough comment on each but you should pick up a book like SPQR by Mary Beard to give you a better rough outline. Then pick your period or theme and dive into more detailed books on the subject. If you prefer audio books the Teaching Company has a surprisingly good set of lectures of Ancient Rome by Garrett Fagan on audible (surprisingly good as the Teaching Company can be hit and miss). I would recommend these as an outline slightly more detailed than SPQR.
How did the city of Rome come to dominate Central Italy? Evidence is low on this time period but there is enough to suggest Rome was a tributary of the Etruscan state. This makes Rome's rise all the stranger, however it is plausible the gallic movement in tribes that lead to gauls settling in the Po Valley and the burning of Rome also weakened the Etruscan cities. Roman sources are keen to play up the military dominance of Rome as the factor to this early prominence but we shouldn't take that at face value. A lot of our sources are written during a time when the Roman army was winning victories across the Mediterranean. From the time it takes for Rome to become preeminent and Rome's ideal location economic factors would appear most likely. For more detail Kathryn Lomas The Rise of Rome.
How did the Roman central Italy regional power dominate the whole of Italy and gain its first fetters of empire in its defeat of Cathage? The tributary system of conquered cities is of prime importance here, keeping states loyal and allowing Rome to continue to field armies despite retreat after retreat. It is the beginning of Rome's ability to co-opt other cultures into a flavour of its own. The change to the maniple military system also allowed the Romans to retreat and preserve their armies in the case of defeat (in most cases a Phalanx that breaks is annihilated). Kathryn Lomas The Rise of Rome also covers this period well.
How is Rome able to dominate the Mediterranean? The military victories in Greece and the East are key. Again the maniple comes to the for but also the professionalisation of Centurions. These experienced NCOs are given much more control of battle actions, allowing the cohorts under their command to act independently depending on their situation without having to wait for orders. You also need to understand the attitude of the Senate and Roman people. To have an empire you actually have to want one. The Romans liked to claim each war was a defensive war and they gained the empire by accident but any examination of the causes of the wars can see through these claims. As one example, after the first Macedonian war is won Rome withdraws its troops, there is no thought of trying to control Greece or even garrison it. However by the second war there has been a change in attitude and Rome is beginning to actively acquire territory as well as more transparent in their client state dealings. For more detail see Robin Waterfield Taken at the Flood: The Roman Conquest of Greece
How did Rome not collapse under the strain this empire exerted on its political system? (that was designed to rule a city not an empire) For this you need to focus on the Gracchi to Augustus in the end of the Republic. This could easily have turned into a similar situation as the crisis of the second century with different commanders establishing their own dictatorship before being ousted in turn. For more detail The Roman Revolution by Ronald Syme.
How did Rome transition from an Empire of predominantly client states with yearly raised armies into a defined universal empire with a standing army? This is all the reforms of Augustus, client states transition, census are taken, land is redistributed. Recent work has shown in North Africa in particular the Roman census and redistribution of confiscated land resulted in a lot of natives (not Romans) gaining. This can be seen as one of the many ways Rome is able to transition using support of native populations. The big factor here is money though, the change of the army to a permanent professional army is extremely expensive. It is no surprise then that we see in the Greenland ice cores that roman industrial activity explodes during this time. The imperial system in the first century AD really cranks up it's gold and silver mining. See Mattingly's, Imperialism, Power and Identity.
That's the extremely rough overview of the chronological phases of the Roman Empire's expansion however there are also themes you should look at. I would recommend looking at the themes once you have the understanding on chronological events.
How was Rome's military able to beat other nations so effectively? Michael Crawford's The Roman Republic covers the period of Rome's military conquests very well whilst also giving you the political context, Robin Waterfield's Taken at the Flood, covers the Greek campaigns in detail.
How did Rome co-opt the ruling classes of conquered nations and resolve resistance/unrest? David Mattingly has done great work on this in his Imperialism, Power and Identity, and in Britain: An Imperial Possession. Also see Louise Revell's Ways of being Roman: Discourses on Identity in the Roman West.
What was it in Roman culture and political system that made them particularly driven in conquest? Michael Crawford's The Roman Republic, covers this aspect also but Jed Atkin's Roman Political Thought, is good for more detail.
What tools of empire and forms of communication did Rome use to govern provinces great distances away from its centre? Martin Pitts and Miguel Versluys Globalisation and the Roman World, covers imperial communication as well as the spread of ideas and discourse in the Empire. I think these all go hand in hand. However there are good essays in the Mattingly books I have already mentioned also.
EDIT: added book recommendations
Too broad of a question for one answer to do it justice. I would recommend reading Polybius’ the Rise of the Roman Empire. It is as close to a first hand account of a historian who does a phenomenal job at creating a clear narrative of how Rome was able to consolidate its power and what elements of Rome enabled it to be so dominant. It’s not an easy read but it is very rewarding if you stick with it and are able to analyze what he writes about.