The first stirrup came in the 2nd century AD from India. From there it spread to China, Timor, Polynesia and as far as Ethiopia. Back then a stirrup was a ring of iron. In the 6th century the stirrup was brought to central Asia, Turkistan, and Iran. Eventually it went to Germany and France were it was adopted by Charles Martel for his army. He saw a means of giving his horseman more stability in the saddle by using the stirrup.
Middle ages: Equitation in the middle ages was rude, sketchy, and designed to put the enemy at a disadvantage. Later on, the heavy armor was not used anymore and the saddle became lighter. The horseman looked for a fast and managable mount. Equitation in the 17th century was developed very elegantly and very slow-paced. Spurs were continually used. Horses were trained around and in between pillars.
In the 18th century the French Minister of War founded five schools of equitation for the cavalry, among them the school of Saumur.
The 7-year war showed that the schools did not work. The Germans beat the French in the war. Fredric the Great created a highly mobile and maneuverable cavalry capable of taking the most uneven and broken terrain and charging the enemy. Voices were raised against the academic equitation but in vain. Germany and France fought about equitation techniques for a long time. Many riding schools taught military equestrianship. The Austrian school, also known as the Spanish Riding school was and is a school to teach equestrianship in a noble form.
In the 19th century the railroad and machine came and horses were not needed for transportation and war. Farms however still used horses for plowing their fields, until tractors became common. Horses were now used more for pleasure and competitions in the equestrian sports.
In 1896 the Olympic games were started up again. In 1908 equestrian games were added in the Olympics. Over time, various sports developed like polo, Fox hunting, Eventing etc.
Rodeo seems to have no formal beginning. It seems to have originated in the days of the open range. Work-weary Cowboys would get together and amuse themselves with feats of horsemanship.