Rome has always had a special place in antiquity. When discussing ancient cities in general Rome must always be discussed separately because it was such a unique case. Its sewage systems were one of a kind and probably one the city's biggest engineering fits, along with its numerous aqueducts.(1) However, to call it 'very advanced' is perhaps a bit of an exaggeration. It was big, sure, but its practical advantages were quite limited.
Until the 2nd half of the first century BCE we have found no evidence of public latrines in Rome.(1) It is likely that many poor people, who had no access to private latrines at their homes, simply defecated in the streets, which in turn were littered with human and animal waste.(2) The streets were periodically cleaned when the various basins throughout the city (Frontinus states that there were 591 in the 1st century AD(4)) overflowed and flushed everything away.(5)

But there were other problems with the sewer system. Many people did not connect their private latrines to the public sewers for very practical reasons: A) Fear of noxious fumes rising from the sewer into people's homes, B) Rodents having an extra entrance into homes, C) Bad smells, D) High costs.(6) Additionally, feces were a valuable source of manure: simply throwing them away would have been a waste. Many people chose to dig pits and collect their feces in there, rather than flushing them down the sewers.

Sewers also require a lot of maintenance to remain in great shape. This was rather expensive. In a lot of cases the Roman empire relied on the beneficence of its social elite to pay for things that were for the public good.(7) It is however unlikely that many people chose to invest in the sewers because they would expect a certain level of social renown to come from their projects. Nobody wanted their name attached to a project that was literally built on shit.(8)

So despite its sewer system, Rome was a relatively filthy city. The large amount of poor people lived together in cramped spaces called insulae. These were high-rise buildings with little to no privacy for the people living on the cheapest (highest) floors.(9) And since Rome was by far the biggest city of its time, approximately 1 million inhabitants, there were a lot of poor people cramped together.
A myriad of factors such as the filth in the streets, overcrowding, endemic malaria and tapeworms, and a huge influx of immigrants, all contributed to the city being exceptionally deadly and actually requiring the large influx of immigrants to keep its population from shrinking.(10) All this made the city a breeding ground for various diseases. This eventually culminated in a deadly event when the Antonine Plague hit the city in 166, killing an estimated 20% of the people.(11)
(My apologies for the digression to ancient plagues. They are my field of research and especially relevant at the moment, so I took the opportunity to weave them into my story.)

To answer the second part of your question: some Romans did indeed have access to taps in their homes. Rome was home to a large number of aqueducts that supplied the city, flowing into 591 basins in the 1st century, as stated by Frontinus. This was usually reserved for the richer people however, and most people would still be forced to collected water by walking to a basin and collecting it, and then storing it at their homes. Despite the accessibility to clean water (at least it was clean when it entered the city) it’s highly unlikely that the water was clean by the time it reached the lips of the citizens. Contamination of basins and storage vessels by the various sources of filth as described above (e.g. carcasses, feces, mosquitos) would not be out of the question. The people who used pipes to deliver water straight to their home were not much better off for a very simple reason. Many pipes were made out of lead.(12) The Romans lead-poisoned themselves through these pipes and certain wine-creation procedures for centuries, leading to higher mortality rates and lower fertility.

So in short: yes, they did have access to advanced sewers and in some cases taps, but it only marginally improved their quality of life.

(1) Davies 2012, 4-5.
(2) Davies 2012, 3.
(3) Davies 2012, 1-2.
(4) Frontinus 78.3.
(5) Davies 2012, 4-5.
(6) Scobie 1986, 411-412.
(7) Zanker 2010, 77 / Kissel 2002, 148.
(8) Davies 2012, 11-12.
(9) Scobie 1989, 427-428 & 430-431.
(10) See Scobie 1989, Scheidel 2003, and Davies 2012 for more in-depth information about the filthy living conditions. See Harper 2017 for malaria and tapeworm
(11) Harper 2017, 115.
(12) Scobie 1986, 423.

Bibliography
Davies, P. J. E. 2012, ‘Pollution, propriety and urbanism in Republican Rome’, in: M. Bradley (ed.) Rome, Pollution and Propriety. Dirt, Disease and Hygiene in the Eternal City from Antiquity to Modernity, New York.
Harper, K. 2017, The Fate of Rome. Climate, disease, & the end of an empire, New Jersey / Oxfordshire.
Kissel, T. 2002, “Road-building as a munus publicum”, in: P. Erdkamp (ed.) The Roman Army and the Economy, Amsterdam.
Scheidel, W. 2003, ‘Germs for Rome’, in: C. Edwards & G. Woolf (eds.) Rome the Cosmopolis, New York.
Scobie, A. 1986, ‘Slums, Sanitation, and Mortality in the Roman World’, Klio. Beiträge zur Alten Geschichte Vol. 68 (68), 399-433. Zanker, P. 2010, “By the emperor, for the people: “popular” architecture in Rome”, in: B. C. Ewald & C. F. Noreña (eds.) The Emperor and Rome: Space, Representation, and Ritual, New York.