Private music instruction is an extremely cost-effective career. All you really need is an instrument, a pencil, and a wristwatch. Other items make it easier, more efficient, and more enjoyable, however!
Just as you consider the impact teaching in the home makes on your family (as well the time it demands and space it takes), also consider your neighbors. This may impact not only the type of teaching you do but also its schedule:
Have a space you call your office. It can be as simple as a briefcase from the thrift shop or a cardboard box hauled out of the closet and set on the kitchen table; or as elaborate as a spare bedroom with real, live office furniture. A closet works well, too, as does a "hidden office" (an armoire or wardrobe which opens to reveal desk surface, filing cabinet, etc.).
Make every effort to be a good neighbor.
Anything you do to prevent problems will repay you handsomely.
Follow any community restrictions on signage, etc.
If your city requires a business permit for a home business, get one. Start at your local city hall. If you plan to sell music at retail (that is you buy at a lower price and sell it at the printed price), make sure your sales and tax permits are in order.
Consider parking if you plan to teach at home. Ask students to park only in front of your house or in your drive, not in front of a neighbor's (surely no one would park in a neighbor's driveway!). If you live in an apartment, are there at least two parking spots for your unit? (One student coming and one going.) If not, you may need to park you own car out of the complex on teaching days so students can use your spot. One of the surest ways to anger your neighbors is to have students park in their assigned spots or across their driveways. Angry neighbors cause business problems, as well as social ones.
Close off the non-public portions of your home to keep students from wandering about. Children, especially, are very nosy and wish to poke in your cabinets, etc. Speak up immediately if this happens. (You'll hear it, you'll later find things disturbed, or another student will "tattle".) Speak to the parent at the second infraction. Consider dismissing the student at the third.
Note: Don't leave dangerous things lying around, such as kitchen knives. Don't leave cash lying about, either, such as a basket of change for busfare.
It's best if students enter and leave through a door you can monitor. This way you know everyone has arrived safely and that no one who shouldn't be there has entered your home.
You'll need access to a bathroom. Consider taking delicate or expensive decorative "pretties" out and replacing them with more substantial items.
Your teaching area should be well-lighted and have adequate ventilation. Since you should keep your windows closed to keep sounds from disturbing neighbors, you may have to add weather-stripping and other sound-deadening materials. During the summer months, you may need an air-conditioning unit for your studio if your home does not have central air-conditioning. No one can concentrate when it is stifling hot - and opening the windows will not endear you to your neighbors!
Hard surfaces - - hardwood floors, undraped windows - - reflect sound and will make your instrument sound brighter and louder. Soft surfaces - - carpet, upholstered furniture, draperies - - absorb sound and will make your piano softer and more muted.