While Yorkshire Terriers are incredibly intelligent dogs that will easily learn tricks and such, it is believed that the Yorkshire Terrier is the hardest to housebreak. Housebreaking is probably the first thing that you want to teach your Yorkshire Terrier, and there is a right way and a wrong way to do this.
The first step is to confine your Yorkshire Terrier to a small space or small room, and to establish a place where it is acceptable for him to relieve himself. It is important that you teach your dog this immediately, and keep him confined until he is consistently using the appropriate area for relieving himself. From there, you can start taking him outdoors to relieve himself. Note that Yorkshire Terriers can be litter box trained as well, but it is important to clean out the litter box each day.
Once your Yorkshire Terrier is housebroken, the real fun can begin. Hopefully, during housebreaking he has learned his or her name. Make sure that you use his name often. You should start training him to obey basic commands; such as sit, lay, come, and stay. Work on just one command at a time, per training session.
Realize that these tiny dogs have tiny attention spans and ten or fifteen minutes of training per session, maybe three times a day is sufficient. The important thing is that you work with the dog each and every day. Do not skip days! Within a week or two, your Yorkshire Terrier should be obeying these commands the majority of the time.
Your Yorkshire Terrier associates words with items as well. Give each of his toys a name, and teach him their names. For example, when you throw a ball, you can command him to get the ball. Don't simply say fetch, and don't throw the ball without saying anything. Use the command 'get' or 'fetch' with the word 'ball.' Do this with all of his toys, one toy at a time, until you can tell him to fetch a certain toy, and he can pick that specific toy out of a group of other toys.
Other than basic commands, house training, and word association, there are numerous tricks that you can teach your Yorkshire Terrier, such as rolling over, dancing, and shaking hands and such, but more importantly, you need to teach your Yorkshire Terrier good manners.
Remember that he will be living in your home, as a member of your family. He will be there when you have guests as well. Your Yorkie must be taught not to jump up on people, and also be taught to stop barking at your command. He should never be allowed to beg for food while you are eating, to eat off of your plate, or to be hand fed while you are eating. Your Yorkie needs to be able to listen to certain commands so that he can get along with the other people and animals in your home.