Hi!
I volunteer/have studied transportation (mostly rail) at a predominant Railroad Museum in the United States! So sorry if you were looking for a more European Historical Answer, but the dates are not that different I promise! (Sorry if you were looking for more European Dates)
EDIT: I found the original video feed that I wanted to use! It is of the 6th Avenue Line in New York City! However, it is from 1916, but it showcases the type of evlvated trains used after New York City's Steam Ban! (Video is the link below)
Rails and trams were some examples of early inter and outer city "public transit"
The first horse drawn trams were popping up in the early 1830's! Brooklyn New York wouldn't see it's first horse drawn tram in the city until 1854! However, many early railroads used horse drawn trams as a means of transportation. The Philadelphia and Columbia Railroad would use trams with horse changing stations in places like Leaman Place, PA. (the entire RR was 82 miles in length!) Horse drawn trams would last up until the early 1850s-1900s when they would be replaced with better, faster electric trams!
Here is a video of Manhattan Streets in 1911 (this is for electric trams)
For railroads, the horse drawn tram would be replaced by the 1840-1850's by the Steam Locomotive! These locomotives would be steam driven up until the 1940s! Railroads would be used to service many local suburbs around the cities! Most of these would also be steam driven in the early years!
We would also see the introduction to above ground steam railways in the 1870s! However, most of these would be torn down in the 1940's-60's due to subways systems becoming more predominant!
This what what they would have looked like!
These would be would be steam driven up until 1900, when New York City would purpose the "No Steam Ban" this would ban all steam locomotives from inside the city limits! Therefore, many of the steam locomotives would be converted to electric locomotives (there is a whole other history here which I would love to discuss!). These trains would end up cutting the total time on local routes in half! Making them much more efficient!
In the end though, many of the public trams would be removed to make way for bus routes, and the growing increase of automobile traffic! in the 1960's and 70's
The idea of road services have been around for awhile! You can see these in coach shops littered around some countries! We can see this from the early 1800s and even before hand!
In the United States one of the most well known is Wells Fargo! These early taxi/buses would haul people across the country and stop to change teams of horses, and do minor repairs and let the passengers get out and stretch!
I love this example of people living the life in a Victorian England Coach Stop
The earliest coach rentals can be found from 1610, it went from Edinburgh to Leith. These of course be replaced by rail services by the 1850s. So by the late 1800s people were relying on trams, and trains to take them throughout the cities!
I do help this answered your question!
Stagecoach East: Stagecoach Days in the East from the Colonial Period to the Civil War by Oliver Holmes and Peter Rohrbach
https://www.youtube.com/watch?v=cyL6wWOHxUM
