how do you really start it up? Do you recommend purchasing a bunch of books to have on hand for each student? Or find out about the students I might get before figuring out what books they need? How do I start the business end of this thing?
Also, what is the best way to be prepared for having new students (and in the home, no less?)
Congratulations on your decision to become a private teacher. Teaching is such a rewarding experience! I'll do my best to answer your questions based on my experience. (I've been teaching privately for 18 years, 14 years of that I was affiliated with a University but continued to teach in my home studio as well)

how do you really start it up?[/b]
First, I would suggest you join your local Music Teacher's Association. There you will find other teachers that can offer suggestions as well as supply you with students. You may also find a mentor whom you can observe and continue to learn. You can also find students by making yourself known at a local music store and asking if you can leave fliers with information about you and your studio. Eventually, if your students are happy with you, word of mouth is the best way to get students.
I would also suggest creating a policy letter of your studio policies, ie: tuition, late fees, cancelled lesssons, make-up lessons, practicing requirements, materials, student/parent/ teacher expectations, performance opportunities, etc. I've found that it is much easier to create studio policies and address certain issues before students begin their lessons. This way, everyone knows what is expected up front. I also have the parents as well as the students sign the policies for my files.

Do you recommend purchasing a bunch of books to have on hand for each student?[/b]
This is not necessary as you are just beginning your studio.

Or find out about the students I might get before figuring out what books they need?[/b]
Yes, determine their needs before purchasing. Each student is different, and there are so many methods out there that I believe appeal to different types of learners. You may want to consider creating a "Music Library" I charge my students a yearly fee of $15. I then have the funds to purchase music to supplement their methods.

How do I start the business end of this thing? [/b]
You're on the right track in creating a business. Ask a lot of questions from experienced teachers. This is a great forum for that.

Also, what is the best way to be prepared for having new students (and in the home, no less?) [/b]
The best way to prepare for new students, IMO, is to know about what is age appropriate to the ages you want to teach, have a solid music foundation, love teaching and children, have a solid studio policy, and plan on having fun!
If you are teaching in your home, it is a must to have a quality instrument to teach with. Next, create a professional teaching environment. (if you have children, get a sitter!) Don't answer the phone while teaching or let other things distract you while teaching. Your student is paying for your full attention.
Also, check to see if you have a Home Owners Association, and if you are allowed to teach in your home. In my area, HOA's are a real problem.
I hope this helps! If you would like me to send you a copy of my studio policy letter, just pm me and I'd be happy to do it.
Good luck!
Cranky
