Were there horse rental agencies, or horse-and-cart rental agencies, in the days just before the automobile came about? I’m aware of taxis, of course, but I’m wondering whether it was possible to train a horse to take commands from a wide variety of riders or drivers, or whether it was more dependable to hire a driver when you hired a cart (which would be the taxi).
There were livery stables, where you could rent a horse either to ride, or along with a buggy or similar horse-drawn transport, but they weren’t at all the trains stations, the way you see car rental agencies at the airports.
Given that urban life, in particular, was not so dependent on private transit in those days, the average tourist or business traveler could probably do just fine with public transportation and the occasional hansom, when it was too far to walk.
To the OP, I have rented horses from stables, and not only were they trained to respond to unfamiliar riders, one at least responded well to a teenager (me) who had never before been on a horse, and another had no problem with carrying my dad with a chihuahua tucked inside his jacket.
In days when joe average might be expected to be an experienced horseman, this would have been even simpler.
Well, what is the English word for those horse farms where teenage girls go to learn horseback-riding? There, horses are shared by often dozens of different, inexperienced, riders, and that seems to go well.
You seem to be confusing “horse training” with “dog obedience training”. Horses are only “trained” in the sense that they are taught that “pull on the left rein” means “go to the left”, “pull on the right rein” means “go to the right”, “pull back on both reins” means “stop”, and “kick”, “hit with whip”, or “make clicking noise” means “go”. And that’s basically it. So anybody can put a saddle and bridle on a horse, or hitch it to a buggy, and make it go somewhere. It’s not like with a dog that maybe won’t accept commands from a stranger. Horses basically Do Not Care.
Hell, there were horse rental agencies in the early 1600’s. Where do you think we got the term Hobson’s Choice? 
Okay, thanks! I figured horse rental places existed, but did not know the name. Livery stable? Is that cognate with ‘delivery’?
If you’re wanting to go horseback riding by the hour, you look in the Yellow Pages under “Stables”. “Livery” mostly gets you limousine rental services nowadays.
I used to go to horse rental agencies, but they were always out of the horse and cart I’d reserved and wanted to put me into a large fancy carriage instead.
Plus you have to watch them like a hawk, or they’ll rent you a horse with withers damage and then claim later that you did it.
In the UK, we generally call those “riding schools”.