I thought MCW's answer had some very good points, and I upvoted it. But I think I will try to expand some more on the evolution of the Roman government "system". (The scare quotes express my doubt that it could be called a system for most of the Western Empire's history.)
To start with, sources are sparse and pretty much all biased, usually in favor of the senatorial class -- the top aristocrats -- because they did most of the writing. By the mid Principate, though, Equestrians -- the much larger rank of rich people who were not Senators -- did a lot of the writing, but even then they were clients of Senators or were expecting to enter the senatorial class as their careers progressed.
For the mid-Republic (say from the First Punic War to the Third Punic War) we have various fragmentary histories and not much more. The late Republic is rich in sources -- Caesar and Cicero, of course plus historians, again -- but they still reflect a limited view of the world. The Principate (Augustus through perhaps Vespasian/Titus) and the Dominate (Domitian through the late Antonines) are pretty well documented and then things get really complicated with the mess of the Crisis of the 3rd Century and then the restoration under Aurelian and Diocletian, which turned to mush not too long after Constantine and the whole thing died a hundred years later.
I wish I could recommend a good book on the evolution of Roman government, but if there is one, I have not found it.
In the mid-Republic, government of the Empire was by the city council of Rome -- the Senate -- and provincial governors typically served for a single year. There was no permanent staff in the provinces, and governors brought members of their own households and maybe a few young men of allied families and made do. The governors, basically, had two duties: act as chief judge in the province, and command any military in their province. (Tax collection was done by private enterprise who bid an amount and then went to squeeze that plus a profit out of the province.)
The best way to understand Republican Rome is if the Mafia ran a country. Everything was custom and legalism (when convenient) and who had the most clout. The late Republic demonstrated this through fifty years of civil wars including Caesar successfully becoming capo de capo, but ending only when Augustus did it with more finesse. It was all very, very personal, and pretty ad hoc.
When Augustus started the Principate, he began an evolution of appointing governors for longer terms -- e.g., Pilate was Procurator (basically, a junior grade governor) of Judea for around ten years -- but does not seem to have done much more than that. By the mid-late Principate we start to see powerful long-term administrators in Rome. (I, Claudius provides a pretty good picture of this, with Pallas and Narcissus and others acting as ministers over parts of the government.) But it was still seen as a Family affair (e.g., Pallas' affair with Nero's mother...)
The early Dominate certainly saw further evolution: Most of the Empire was run by legates appointed by the Emperor and their terms of office were often long enough for them to develop a process, but it was still very, very personal. The best example of this is the letters of Pliny the Younger, a senator who was in Trajan's second circle of buddies. Pliny was governor of Bithynia and published his correspondence with Trajan and it has survived. Pliny writes many dozens of letters to Trajan asking for detailed instructions on how to do this or that, and Trajan responds personally. (He had a secretarial staff, and probably dictated parts and had the staff fill in boilerplate for others, but still. There were dozens of governors at Pliny's level, all wanting the Emperor's ear. Overwork was the Emperor's constant companion.) The picture we get is still of an Empire which has very, very little routine bureaucracy.
The post-Constantine Empire had developed something like a bureaucracy. We have an excellent picture of the state of affairs around 400 AD in the Notitia Dignitatum, a work describing a couple thousand civil and military offices around the Empire. But this organization was too little, too late and the Western Empire fell 75 years later.
Some general comments:
First, I've ignored the military, and this is probably a mistake. While the war-fighting function of the military is probably irrelevant to this question, there is evidence that in times of peace, the army did a lot of miscellaneous work in the provinces, especially what we'd call public works. They also probably provided police functions, to the extent that such existed. I suspect that a lot of it was informally organized between the governor and the top army officers in the province.
Second, the Empire was an intensely urban thing -- even near Rome, the countryside could be lawless and Rome didn't care a whole lot about it. Rome relied on its cities to provide local services for each city and its hinterland. (Service as a city magistrate brought Roman citizenship and other Imperial bennies.) Rome really didn't want to mess about running cities and as long as they were good citizens (i.e., paid their taxes and didn't revolt too much) it usually didn't care.
Third, it's very important to understand the Roman familia. This not our nuclear family, but would include dependent clients and a large number of slaves and freedmen. All were people who owed support to the head of the familia. I'm still not sure I really understand the relationship, but it seems, more than anything, to be very Mafia-like. But everyone from the Emperor down used their familia as their staff. (Which meant that continuity was not terribly great.)
Fourth, for most of the time, Rome did not have a written law code. It had laws, to be sure, and some were recorded, but it was all very messy and complicated. In the high Republic, when a Praetor (essentially a top judge) took office, he would publish a list of the laws, procedures and customs he intended to follow. If he messed about too much, of course he'd get push-back, but it was still very much a personal thing.
So, that's a broad outline of Roman government. It's only that, but I think the general picture of an always-evolving, terribly ad hoc, very personal approach to government is accurate.
I wish I knew enough about ancient China and could make a comparison. I hope someone does.
I'm currently reading Mary Beard's new book Emperor of Rome (an excellent book which goes into a lot of detail about Roman administration). In it she writes:
Leaving aside the army, no other empire in history of the world has operated with so few official boots on the ground (the Chinese empire had proportionately twenty times the number of senior administrators than did Rome).
pp 232-233. This is not original research, but Beard certainly counts as a pretty authoritative source.
Added Even Later:
If you can find a copy, Ronald Symes' 1939 book The Roman Revolution goes into the development of the bureaucracy and government by Augustus in considerable detail (chapters 25-27, particularly).
It's consistent with what I wrote, above.