I have written these mounting instruction in steps so that it’s easy to figure out where you might have a problem. However, the whole action of mounting should be executed smoothly and effortlessly, as it's one continuous motion. You can view the pictures to help you see if this is how your mounting looks. (I'm using a Western saddle in the photos, but the steps are the same if you are using an English saddle).
STEP 1: Hold the reins in your left hand and grab some mane just in front of the pommel, (front part of saddle) now place your right hand on the cantle (back part of saddle).
STEP 2:Place your left foot into the stirrup but not all the way in. You will need to keep your knee tight up to the saddle, as close as you can get it. You are facing forward, slightly on a diagonal to the opposite shoulder. Some people like to start facing toward the back of the horse and bounce three times until facing forward but this usually ends up with the toe of the boot kicking the horse and causing him to move.
STEP 3: Push off of your right foot, immediately straighten your left leg, and get centered over the horse. Your whole body will be straight at this point. Your right hand, which had been on the cantle, will move to the pommel on the right side of the saddle, and both hands will now be holding most of your weight — very little will be in your left foot, which is in the stirrup. This is the place where I see people making the most mistakes, they are focused on bring their leg around at this point and never brought their hips over the center of the horse. If the saddle is going to slip, this will be where it’s going to happen, if done wrong.
STEP 4: If your horse continues to stand still, then you will bring your right leg around the back side of the horse making sure to clear the horse's rump and the cantle of the saddle. Note that if you kick him with your boot, then he might walk off.
STEP 5: At this point, you are still holding your weight mostly in your hands and, as you get ready to settle down into the saddle, your thigh and knee on your right leg should touch the saddle first. Transfer your weight from your hands to your left leg, then fully weight your seat. You will then place your right foot in the stirrup. If you are not able to get your foot into the stirrup without leaning over, please do not do this while the horse is moving! Your horse should continue to be holding perfectly still, but if it has moved off, then ask it to stop before trying to get your foot into the stirrup — leaning over and looking down while your horse is moving off could get you into trouble.
STEP 6: Now that you are mounted, you should have your horse wait for a bit before asking it to leave. This lets your horse know that just because you are on does not mean you are ready to go.
How To: Safely Dismount from a Horse
When dismounting, you simply do it in reverse. The only thing different is that if you used a mounting block to get on, you will not jump back down onto one when getting off. You will pull your left foot out of the stirrup while laying your hips on the saddle, then push away and drop down to the ground. If you are tall enough to have one foot in the stirrup and one foot on the ground without any strain on the stirrup, then you can do that. I do find that people land on the ground and then jerk the saddle sideways quickly, which is problematic for the horse and his back.