In order to be a successful professional photographer you need to be as good if not better in your marketing as you are at making images. The market is so heavily saturated with good photographers you need to be on your A game when it comes to the business side. As others have said Instagram is a great tool, but it's not as easy to promote yourself on there as it used to be. I would work on networking within your market. Word of mouth referrals can be a huge part of your business.
Start by getting a super good quality camera, if you have that already get an Instagram and just post. Post your pictures everyday but make sure they are quality photos. Go to horse shows. Go to a local Dressage show, local Hunter/Jumper show, even cross country and the race track. It keeps your feed and photos interesting.
Hope this helps!
Also put posters up at shows in the bathrooms or on a notice board so people know your details. Also wear a specific colour so people think of it's the photographer in the red shirt here again today so you become more of an icon rather than just a photographer
A great way to start is going to shows and taking pictures of the riders, and posting them on Facebook. You can layer add watermarks and sell them for a commission. People will eventually start waiting for your pictures and following your company's page. You can then advertise things like photo shoots and gradually increase your prices