Alot of people do it differently but I normally starting lounging them on a 30 foot rope. They run in circles around me while I hold the rope and I tell them commands.....such as walk and make a clicking sound, or just tap them with a whip. To make them trot, I click again or snap the whip behing their butt (NEVER hit them with it, just a very gentle tap. The whip is a training aid and not to be used for punishment. A horse should not be made to fear the whip.) To make them slow down, I gently pull on the rope or put the whip toward the front of their body. Once they get to know the commands, which takes about a week for them to get to really know what I want, then I normally put the saddle on them.
Today was a little windy, so I didn't do the lounging part. I just brushed her, then rubbed the saddle blanket on her and when she accepted that, I held out the saddle for her to smell and then gently put it on her back. I strapped it on just tight enough for her to get to know what it feels like and to keep it in place in case she went crazy.
Once they feel comfortable with the saddle when they walk, trot, and lope, then I normally put them in a small area or have someone hold the rope...stand on a bucket and lean over the saddle. If they are ok with me doing that a few times I will try to gently swing my leg over the saddle and hang on! LOL
Once I'm actually in the saddle and the horse is ok with it I have the person walk the horse a few steps. Some horses get a little nervous because the weight of the person throws the horse's balance off a little. It took a full week for one horse to learn how to lope with me riding her.
All horses are different. One horse I had broke within 2 weeks and I felt confident to ride the trails with her, another took me over a month before I would take her outside of a fenced in area. It all depends on how much they trust you, what their personality is like and such.