Horse riding is a vigorous activity. The fitter you are the more stamina you will have to enjoy life in the saddle without sore muscles or getting tired afterwards. Of course, there are rider disciplines which will require a higher level of fitness than casual riders. Eventing, polo, or classical dressage are good examples where more stamina will give you an advantage.
Those who take the time to develop their stamina and fitness will enjoy horse riding at a higher level. Studies have also shown that fit riders are less likely to experience a fall.
In the first few weeks of riding, a once weekly ride will be sufficient for you to develop the muscles used for holding a good posture. It's essential you don't develop any bad habits in the early stages, which is why training with a professional instructor at an approved riding school is highly recommended.
Cardio at the gym or running will improve your muscle strength and stamina for horse riding. Excessive strength training will increase your muscle mass, which may be detrimental to efficiently riding a horse. Fortunately, building muscle takes a lot of effort and strict dieting.
What you should be aiming for in your exercise is to improve the flexibility and suppleness in your joints, which will give you greater freedom of movement and better control of the horse.
A training program that builds strength and suppleness in your lower back and stomach will benefit your riding and enhance your stamina.
Other areas to focus on include your hips, thighs, and calves. Hip joints are incredibly important in riding. Strong, supple, and flexible hips will give you freedom of movement you need to control the horse and maintain poise during jumping and flatwork.
Your lower back needs to be strong because it's integral to controlling the horse's movements. If you don't have adequate strength in your back muscles, then you are likely to tire quickly, and may experience some discomfort after a ride.
Consulting with a professional fitness trainer is the most efficient and safest way to ensure your routine has the best exercises for you to achieve your horse riding goals.