Here is a fashion plate from Godey's fashion Pages. This gives you an idea of what some women wore for horseback riding. Gradually womens riding habits changed, not hanging so low to the ground. Women also were taken to wearing leather breeches underneath their riding habits.
Transportation has long since changed since the horse and buggy.
Back in the days of ole if one did not have a horse he simply walked. And in town he could always hire a cab or in bigger cities he could ride what appeared to be a horse pulled trolley with windows. Stage coaches could carry a few people along with luggage. Still, the real hero was the horse. Since it took such a fine and noble animal to pull such weight. The horse is what made the carriage ride a romantic and endearing ride. Maybe we can't go back in time. But you can always dream about it.