Geralt always rode on a horse called 'Radish.' This was the most loyal horse a man could possibly ask for. No matter where Radish was, it would trot up to Geralt at a single whistle.
Cirilla and Olivia rode on another horse. Geralt was highly protective of his daughters. He would use the 'Axii Sign' [1] to keep the wild horse under control whenever they were on one. That way, the horse would not be liable to buck his daughters off suddenly.
That's right. Geralt was rather poor. He could not buy them a horse. The most he could do was capture some wild horse at random and have his daughters ride it.
Geralt was exasperated. His adventures paid a fair bit of coin, so where did all the money go? Was he too addicted to the Gwent playing cards? But he wasn't he a master of that game?!
