May I go horseback riding on Chol Hamoed & may I take training classes for horseback riding on Chol Hamoed as part of the course that I am taking of which one of the training days falls on Chol Hamoed?

Yes. It is permitted to go horseback riding on Chol Hamoed even if it is not necessary for the need of the Moed. Hence, certainly you may do so as part of a Training course which would otherwise cause you a loss if you were not to participate.

Sources: See Michaber and Rama 536:1; Beir Moshe 7:13