What you describe is a rather typical big city in Europe or Asia. The majority of their dwellers do not use personal cars on an everyday basis. Public transportation is usually a faster, cheaper, and more convenient way to move around the city. In Tokyo, a trip that takes 30 min by train can take an hour by car (never take a taxi if you are running late for your plane! look for the next train).
Considering this, some comments on the suggested differences:
Cities are denser and tend to built up and down (underground) rather than spread out
It depends on the available transportation and its design. Too high density may also lead to problems: Public transportation can transport only so many people. A station that is fine with 1 mln passengers per day may not function well if it has to handle 2 mln. The busiest station in the world is Shinjuku station with its 3.6 mln passengers per day. It is a major transportation hub and it tends to become very crowded during the rush hours. I am not sure how many more passengers it can handle without a major redesign.
People don't tend to move around as often as they would in our world and adult children likely don't move far from their parents/family
I am not so sure about that. Americans do move more often than people in other countries. But it seems that the reasons for this are a bit more complex than personal vehicles.
Learning to drive is a much stricter, regimented affair than it is in our world. Rather than learning from family or friends like most of us do, people who drive have to driving school
The USA is the exception, not the rule. It is very famous for its relaxed approach. Most developed countries require formal education and have much more challenging driving tests.
Roads would be narrower to accommodate the lack of cars
This depends on commercial traffic. If goods are still transported by cars the roads may have to stay wide. Please also note that many European and Asian cities do not have wide roads to begin with.
Foot police would become commonplace, walking amongst the crowds that are commonplace in cities
Maybe. Or maybe they will switch to smaller vehicles.
Public transportation (bus, train, tram) would be ubiquitous and would have to be very sophisticated to handle to amount of people using them daily
This is a definite yes. And we have already existing examples: Tokyo, London, or Moscow. All these cities have very developed and sophisticated public transportation. There is no real need to own an automobile if you live in one of them.
Moving services would be very common and fairly well paid
Isn't it the case in the US already?
Skybridges would be commonplace for people to walk around the city without being exposed to the elements
Elements are not that much of a problem in a big city with a sophisticated public transportation system. At most people have to walk 5-10 min to the nearest bus/tram/trolley stop or train/underground/metro station. City dwellers spend most of their time indoors cars or no cars.
Some additional suggestions for changes compared to the contemporary US:
The US favours separate-use zoning where each zone can have buildings serving only one function (residential, commercial, industrial, etc.). This is done for many reasons some practical, some political. These zoning patterns are not very suitable for big cities and often lead to phenomena like food deserts.
Mixed-use development is a traditional pattern of city growth. The main advantage of this approach is that everything a city dweller may need can be found within a walking distance. This helps to avoid problems like food deserts and at the same time lessen the pressure on the transportation system: There is no need to travel far to buy groceries or other daily necessities. An additional benefit is that food can be bought daily which may lead to a better diet both in quality and variety.
The US is behind the rest of the developed countries when it comes to fast trains. There is only one route that exists today: Amtrak's Acela connecting Washington, D.C. and Boston. According to Wikipedia, Acela trains are the fastest in the Americas, attaining 150 mph (240 km/h) (qualifying as high-speed rail), but only on 33.9 miles (54.6 km) of the 457-mile (735 km) route.
HSR is the fastest type of ground transportation that has the potential to connect more areas than automobiles can. If HSR is highly developed cities can expand their land territory very far. HSR also makes visiting other cities and areas a much more convenient and pleasant experience.