On the Velika Aleja in Ilidza you can take a fiaker to Vrelo Bosne. It is good fun. But actually I recommend to take the walk to Vrelo Bosne and then take the fiaker back. Some fiaker have one horse and others have two.
Going to Vrelo Bosne I think I heard the price of 10 KM per person. Coming back we paid 15 KM for the two of us.
I recommend that you visit Ilidza on a weekend. It may be much more crowded by local people but this also makes it way more interesting.
Anyone will know the Velika Aleja, this old tree-lined alley, that dates back to the Austro-Hungarian times. The alley is depending on the source 3,5 to 4 km long. People walk along to reach Vrelo Bosne, the source of the Bosna river. But not only there are people walking but biking or jogging and on weekends there horse-drawn fiaker carriages will run too.
We walked up to Vrelo Bosne and took the fiaker back.
From downtown Sarajevo you may take tram number 3 to Ilidza. From Sarajevo railway station it's number 4.
The journey costs 1.80 KM if you buy the ticket from the driver or 1.60 KM if you buy it from a kiosk. Please note that you need to validate your ticket when entering the tram. Take an entrance marked "ulaz" as there are not many validating machines in the tram which is no problem if you enter at the end station but if the tram is full well then you will just have to make your way to the machine somehow.
The journey takes quite a while to Ilidza, I'd say at least half an hour.
At the beginning of the Velika Aleja there is a big fiaker parking lot. Don't worry, across the street you find a parking area for your car if you absolutely have to drive here.
Part of Ilidza, Banja Ilidza, has a history as a spa town. You will see some hotels called "Austria" and "Hercegovina" in the spa area that are dating back to the Austro-Hungarian times.
If I got it right, then the heir to the Austro-Hungarian throne, archduke Franz Ferdinand and his wife Sophia, who both got assassinated at the Latin Bridge in Sarajevo stayed here in the hotel "Austria".
Currently, the fountain on the square between the hotels is refurbished.