There are two menus: Tricks and Training. Click either of these and you will be given a selection of commands. To learn how to perform one of these commands you must click on it, watch the dog and then rate it good or bad. If the dog doesn't perform the trick you should click bad. If the dog performs the trick, select good.
The only way to learn these commands is through repetition. If you continue to repeat this rating process, all the hearts will be filled with pink and a notification will flash up on the screen saying "_____ was learned"; your dog has now learned the new trick.
You will start off with only three tricks available in the 'tricks' menu. You will receive more options as the game continues. You should start off learning commands from the 'Training' menu as you will be asked to perform a specific command at the Pretty Dog Contest.