On your pet's page when it is not busy doing something there will be a train button under their name; provided they are not at a higher level than what you can train to with the level of riding hall you have. The level of your riding hall will determine how high you can train your pet and how many pets you can train at the same time.
Level 1 riding hall will allow you to train a pet to level 10 and up to 3 pets at the same time. Each additional level of riding hall increases the level you can train to by 10 and opens up additional training slots for more pets (to a maximum of 20 pets at a time).