After you've completed the preliminary training to start your Season as a jockey return to the Academy and make a 100 on those three tests. After you score a perfect 100 on the three Practical Tests a new Test for Abilities will open up and give you a chance to practice your jockey abilities.
New horse for no points
If you do not have enough points to buy a new horse, select "Next Month" then negotiate for a horse. If there is not a horse that costs 0 points, select "Next Month" again and negotiate for a horse that costs 0 points.