With your own horses, you can breed straight from your Stable. If you wish for a quick view of your horses stats before breeding, you can go to Stable -> Estate Management -> Breeding Management.
To breed, horses must be fed and boarded. From the breeding grounds, select the horses you want to breed in the dropdown menus and press the "Breed". You will either have a success or a failure and have to wait for a cooldown.
Make sure that you do not breed two Overo's. The gene is Frfr in the game. This will lead to the foal dying, which is normal due to the lethal white syndrome in horses. You must breed one that has frfr, which means it does not have the overo gene present. Tobiano and Overo are perfectly fine to breed, but a Tovero is still not possible to breed with another Overo.
That are times that players, and yourself, will put up a stallion for stud and they can breed after paying the fee.
You can breed straight away from the horse's page on any horse up for stud whether it be someone else's or yours from the Breeding tab on a horse's page.
Eqcetera offers a 1/2 stud fee refund for failed breedings to studs. Right now, we currently do not have an active timer of how long they have until they can breed again.