Two ways your work is done for you already
1) To an extent the difference between Europe and the USA already suggests some of what the difference would be. Of course even western Germany and France are still quite choked with automobiles and they do have a massive investment in autobahns/autoroutes. Still it was very striking to me, as an American, to take a road trip from West Germany (this was 1990) to Paris. The French followed or paralleled (or inspired) a suggestion by NY Senator Monyihan that freeways should connect big cities but shouldn't go from core to core but rather sweep through undeveloped land as much as possible and that there should not be new development along them, so they remain clear high-speed routes and not avenues of new urbanization. (IIRC the French Autoroutes are also toll roads which may help discourage new sprawl along them). So, the drive through France was through open country and the approach to Paris was amazingly abrupt from the point of view of American expectations. Even quite close to Paris, it was still open, bucolic countryside (except for the weird colored dots mysteriously placed on the side of the road, presumably some kind of art project). Then there was some sort of large, weird-looking building, probably also something arty. Only then did one see the remarkably sharply-defined edge of Paris itself looming on the horizon; there was no sprawl of gradually increasing development to herald approach to one of the greatest cities of Europe. Inside Paris was nothing like an auto-free utopia of course; we never took the Metro because I was traveling with a disabled person who needed a wheelchair so I can't compare but there was enough surface auto traffic to make it look like New York, driving at alarming speeds and rather maniacally too, was my impression.
In general, flying over continental Europe and looking down, the pattern seemed radically different from America, where even in quite rural areas one sees an extensive sprawl of development and more or less urbanized areas spread from horizon to horizon. In Europe the towns clustered in tight, leaving large tracts of either apparently undisturbed forest or neatly cultivated farmland without any clutter of dwellings or anything industrial. I only ever saw England a few times and there I got a more American impression, so actually I wonder if this is more of a cultural divide between the Anglosphere and continental society. Say the English valued country living and regarded cities as places in which one scrambled to make a fortune but having achieved it, one secured a respectable place in the country and merely used the cities for convenience; this would explain the explosively fractal pattern of dispersed settlement and certainly carries over to American attitudes. Whereas in France to live in Paris itself is deemed the goal of all worthily civilized people and perhaps in general there is a carryover of the Roman ideal of urbanization equals civilization (as the very word suggests!)
Still, despite the bustling and well-accomodated presence of lots of cars, Western Europe in general is much closer to the sort of patterns that would be mandated by a ban on private autos or a decision not to invest in road development. With housing tightly clustered, with public transport in various forms available, with a dense network of rail heavily used for long-distance transport, Europe is halfway there despite all the autos about.
2) There is a series of books dubbed, after the title of the first one, the "Foreigner" stories, by C. J. Cherryh, about an alien world onto which some human colonists (who had intended to colonize some empty world, but they got lost) settled, but ultimately only on a small island because the mainland was dominated by the native intelligent species, the atevi. When first found the atevi had only an 18th century level technology, but they had some other advantages and humans and atevi turned out to have some deep psychological incompatibilities. When the story opens it is a few hundred years later, and there is one man who has the job of serving as human ambassador to the atevi domain; his job title is not so much "ambassador" as "translator" because mutual communication is tricky. Another aspect of his job is to slowly dole out human technical knowledge to the atevi regime, being very careful to permit only development along lines that do not threaten the more advanced but overwhelmingly outnumbered human minority on their isolated island. By now the process is along to roughly equivalent to sometime between 1900 and 1950, in various different technologies.
And in the matter of transport technologies, the humans and the atevi leadership saw eye-to-eye in the wisdom of preferring a controllable, public-utility model for long-distance mass transport and have invested heavily in railroads and not at all in roads for automotive purposes. I won't go further into it because the Foreigner stories are clearly off topic for the forum, but in the course of those books you get a lot of looks at what such an alternate network might look like.
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Now in general--there will of course be roads. Roads of some kind existed for thousands of years, on their simplest level they are simply tracks left by frequent choice of this overland route by foot--human feet, horse feet, oxen--of course any wheeled vehicles tend to define the tracks much more distinctly!
It's just that our non-automotive society doesn't choose to invest in improving the roads much. Development implies more intensive build-up of railroads on various scales and not improvement of surface roads, tighter urban development and much less sprawl into the countryside. Still there would have to be some degree of automotion, in the form of basically off-road vehicles that would be heavier and slower. But going more slowly might offset the structural heaviness of such rugged vehicles, because despite the greater friction lower speeds mean less power demand and hence lighter engines. Electric vehicles would be heavy but appealing in their simplicity and robust operation. Steam engines might be appealing in their straightforward evolution from railroad locomotives and their robustness. Stirling engines would also have their appeal. All of this serves minority, specialized needs of people who must perforce live and work in rural settings; policy favors not them but the much more concentrated urban population and deliberately seeks to prevent the appeal of individually operated vehicles from transforming the urban transport picture.
In the towns and cities, there must be effective public transport. Not necessarily publicly owned though that would be a strong likelihood, but public in the sense that a vehicle you don't own is available to take you where you want to go when you want to, and operated not by you but by a professional driver. This favors tighter settlement patterns, to shorten the routes and concentrate the pickup/drop-off points so that a reasonable number of people show up to be taken aboard during a vehicle's circuit, to justify their operations.
Experience shows that it is not enough to merely have good public transport options; even such American cities as Los Angeles had this and yet the private auto choked off the intercity Red Cars and the local trollies that once fed them. (This did not happen merely automatically; various agencies did conspire to deliberately kill off the Red Cars for instance as proven in a public court trial in the 1940s or '50s, but the fact that the plantiffs in that civil trial were awarded $1 in damages indicates how this corporate conspiracy was indeed moving with and merely facilitating the trend of the times; had it been "against the public interest" in the sense that numerous and powerful publics were harmed surely the judge would have taken it all more seriously and indeed the conspiracy would never have gotten off the ground). Other American cities did retain effective public transport, still used extensively by a broad (in some cities, overwhelmingly the majority) sector of the public. Yet their streets are also choked with cars. Even in New York, the cars might mostly be taxis, but they fill the streets despite the availability of cheaper modes like buses and subways. Even in Europe where the private auto is less favored by policy and infrastructural investment, there are lots of cars and cities like Paris are again filled with them.
So there has to be a strong, deliberate policy decision to forbid private cars, frown on taxis, replace buses with trolleys wherever possible, and so forth. If private autos are merely discouraged with heavy fees, taxes, and the like, they will be seen as signs of great privilege and prestige and people will seek to acquire them despite the barriers--then work as a growing (and especially influential) faction to bring down the barriers, both regulatory and infrastructural. So private cars have to be kept strongly in check.
People like them because they are available to take oneself where one wants to go from wherever one wants to start, whenever one likes. And of course one can leave lots of stuff in them, so they serve as a mobile locker. A public transport system has to address all these issues--has to be kept going on at least a skeleton basis 24 hours, all days, even holidays. Has to be ubiquitous enough to reach even more obscure people, and has to be efficient enough to get people from their homes to their workplace to their places of shopping (and get the goods they buy home) and leisure and then back home again at whatever odd hour of the night. (And provide these services for people who work odd hours, or the society will be unduly restricted in being able to operate flexible workshifts).
With all that investment, can it be done? I suspect so, at least in a suitably alternate history. At some point a society that very strongly values the alternative has to emerge. Presumably this would not happen by universal consensus, though it could easily be by majority consensus. Someone would be pointing out the drawbacks and loudly demanding freedom to do otherwise; there would have to be strong public reason to not grant these demands.
Would it be inefficient? In some respects, maybe. Certainly I just pointed out ways in which it would be hard to accommodate certain needs.
However, rail transport is in mechanical terms more efficient than road, due to the low rolling resistance. Also one can load quite a lot of weight onto a good railroad car so it can be quite dense. Grouping payload into batches tends to be more efficient (offset by the need to wait for a worthwhile load to accumulate, so the first arrivals are idling around waiting for the last ones). For these reasons rail generally works out to be much more efficient in terms of power per ton/mile. Transferring between different vehicles to eventually get near one's ultimate destination (and of course needing some yet other mode of transport to get to the first pickup spot and then from the drop-off to the actual address one is going to) is inherently inefficient and brings delays relative to just getting into one's own vehicle and going directly to the ultimate destination. However the more intense and elaborate the public system is, the more these delays can be minimized and the more direct the route via the public system can be; meanwhile a highly developed public system can include segments that are much faster than an equivalent automobile route can be and avoid gridlock too by planned coordination of the traffic (and because traffic gets aggregated into batches, so there are fewer vehicles to conflict, which also simplifies the route planning and orchestration). Again efficiencies inherent in more concentrated payload in fewer, professionally operated and maintained vehicles moving on dedicated routes come into play also.
The upshot should be, that overall transport costs are lower. Considering that the public vehicles need paid, professional drivers the costs might appear to be higher, but private transport only appears to avoid that cost--society suffers opportunity costs as working people must for a time operate road vehicles "for free." If they didn't have to they'd be freer to either enjoy the time they are being transported by someone else, or get some homework done for their jobs. Considering that clearly such a society is overall spending less on transport despite the intense and elaborate public investment. And the investment being public, it is subject to deliberate modification in response to changing conditions.
I can see two types of society going this route. One would be a radical socialist democracy. The other would be a strongly anti-democratic oligarchy. Either might take shortcuts away from the ideal I have outlined, the latter would be quite likely to. But either can simply mandate a cheaper, less effective, system and then call it good, suffering costs in restriction of options (and probably some degree of public grumbling) that they judge they can bear well enough. The oligarchy might well permit private transport (actually probably chauffeured) for the privileged few and tell the rabble to just shut up if they know what is good for them. Sad to say this is exactly the sort of thing that happened in the Soviet bloc so if some people choose to see no distinction between a radical socialist regime and the corrupt oligarchy, it is that much harder for me to argue with them. (But I would anyway--it would just be harder!)