You can check the progress of the mating by looking under the stats of the female horse. There you will see how much more time is left until the mating is done. Once it is done, you will see another button appear when you highlight the female. Clicking this will prompt you to name your new foal and allow you to see what tier, pattern/color, and gender it is.
So to sum up the steps:
1.Have a male and a female horse.
2.Both horses must be in the same stable.
3.Both horses must have max stamina.
4.Register your male horse for mating at the minimum price.
5.Select your male horse.
6.Open the rent-able mating horse window and highlight the female horse.
7.Wait the given amount of time for the process to finish.
8.Acquire your new horse.
2b. With other people's horses
I think this is self-explanatory as it's pretty much the same way you'd do it with your own male horse, except you don't need to go through the process of registering the horse. Just pick a horse of your choice that you want to breed with and pay the amount required. However, unlike breeding with your own horse, you won't get your money back.
3. Informal Breeding
I call this informal breeding because you can force a horse with no more breeding attempts to mate once more at a heavy cost. This was introduced not too long ago into the game and is a quick and easy way to get a new foal technically without the wait time. As with formal breeding, you must make sure that both female and male are in the same stable. Both horses don't need full stamina for this. However, one horse MUST have at least 1 breeding attempt left otherwise the method won't work. The best time to do this would be when a female has [0/1] breeding attempts and a male has [1/2] breeding attempts left, that way you can maximize the amount of horses you can create. The cost to do this is always 35,000 silver.
WARNING: Doing this will result in the removal of both the male and female horse from existence. Only do this if you really need stable space and don't mind losing the horses you are mating.