Whenever you win a bigger competition you need to break in your horse to the next level so that you can use them in upcoming competitions. Before you can also enter the local competitions to need to revisit the trainers on the Beach, in the plans, and in the forest, you farm and train on the next levels there. Once you complete that you will be able to take part in the next set of local competitions.
Before you can do the small local Competitions you need to unlock new levels in your training in the Forest, Beach, Field and/or the horse barn. Once you unlocked those you can compete in local Competitions.
You Can find local competitions on the Beach, in the Forest and on the Plains.
The Beach and Plains Local Competitions sights are pretty easy to find but the forest on is a bit difficult, once you enter the forest you can head up like the way you would go to get to the field where the pond is just when you get there cross it and the river and head down that small forest path that will lead you to a house that is hidden and that is where you can do Forest Local Competitions.