I will definitely use horses for transport in SHTF. They make less noise than a motorbike; can even swim through a river. Can carry a load or children. Can warn you if someone is coming. Can run very fast.

You will need proper saddle’s and maintain the tack very well to prevent saddle sores. ( In South African wars, most horses died from bad saddles and the English brought in their own Thoroughbred’s that went lame very quickly)
Horses need 45 liters of water per day. Horses need to eat every four hours. In South Africa you need at least 10 hectares natural grazing per horse – no extra concentrates.
Some breeds will be able to survive harsh conditions easier. Need to look at a breed that has got hard hooves, stay fat, calm temperament, not to big to be able to travel underneath trees/ eat less. If possible from local area and already build up immunity to local diseases; used to local temperatures. I would maybe keep at least two horses. One to ride and one as pack horses. They might call out less having company.

You can also get much closer to wild animals being on a horseback. Having experience with horses before SHTF would be a great advantage.