My camera has a sports setting that allows me to take quick action shots, it's how I get all my riding pictures! I just hold the button down and it takes pictures in a burst until I let up. Then I have a few different stages of the moment I was trying to capture and I delete what I don't like. (For example: I would have a picture of the horse approaching the jump, taking off, over the jump, landing, and pulling away from the jump.)
I think, the three keywords are: patience, observation, responsiveness. Just patience and perseverance are useful for take pictures. We live in a world where everything goes very fast! A short question that leads to long speeches! You have to surround yourself with good people who will advise you on photography and how to improve ...
Personally, I have a Canon 1Dx et 5D MkIII, my lens are canon (70-200 2.8) and sigma art (35mm 1.4, et 12-24). Actually, I realized with the time, I use frequently the same lens and keep only this trio.
I hope this helps, bye :)