1. The age and readiness of your child.
I recommend pianists be at least 4 years old and string players be at least 6. All budding musicians should know the alphabet through G, the difference between left and right, numbers through 10, and preferably, how to read simple sentences.
2. You need an instrument.
I very much enjoy helping my students procure an instrument for studies, so I'm happy to help explain the process to you and make recommendations according to your budget. I do recommending renting an instrument for the first few years to see if your child will stick with music before you invest in an instrument. Pianos can be rented for $50-75/month, violins and violas for $20-30/month, and cellos for $55-70/month.
3. Be prepared for the cost.
The cost of lessons is ongoing, as you know. Expect to spend $50-100/year on books and accessories, recital fees (usuall $15-20 per recital for each child), and additional costs if your child takes supplemental classes or participates in an ensemble.
Instruments that are appropriate for private study are of a slightly higher caliber than those used for school instrument programs. Expect to spend at least $1500-2000 on an entry level acoustic piano (digital pianos are not acceptable), $600-1000 on an entry level violin or viola and $1500-2000 on an entry level cello. As your child progress, it is necessary to upgrade to higher quality instruments.
4. Be fully committed to practicing and attending regular lessons.
If you just want your child to be "exposed to music", private lessons are likely not the right solution for you. Many communities offer group music classes well suited to this purpose.
Private music lessons are not an activity that can be successful without daily practice. It is not sufficient to simply show up for lessons and not do any work outside of the lesson time. You and your child must understand the importance of daily practice before committing to lessons.
5. Understand that it is your responsibility to make sure your child practices.
Many parents want to teach their children to be responsible for their activities starting at a given age. With music lessons, it's different. It's not reasonable to expect most children to practice without your guidance, encouragement and enforcement. You will need to ensure that your child has enough time to practice daily and that the assignments given in their practice notebook are completed from lesson to lesson. If you have a hard time with this concept, ask yourself if you would have practiced at your child's age without being told to? Most parents say no. If you did or your child does practice without being told, count your lucky stars! You are truly one in a million!
6. Minimize other activities.
It's important to ensure your child is not so busy that they actually can't practice each day. Beginners will spend only about 10-15 minutes each day practicing. But if homework, family activities, sports, play time and other commitments prevent the child from having sufficient time to practice, not much progress will be made. I recommend children only participate in only one sport at a time in addition to music lessons.
7. Kids who practice like lessons better.
It's commonly assumend that kids who like lessons will practice more. This is not actually the case. Kids who practice and experience regular success and progress like lessons better and will practice more.
8. Set up a practice-friendly area in your home.
Practice areas should be free from distraction. This area shouldn't be in a room with a television, in a high-traffic area of the home, or set aside in a lonely corner that promotes isolation. It should be somewhere the child walks by frequently, so that the instrument has a constant presence in the child's life. If your child is a string player, invest in an instrument stand so they don't have to take the instrument in and out of its case to practice each time. Keep other children and animals free from the practice area to encourage the child's best level of concentration.
9. Tell your teacher about your child.
It's important to let your teacher know of any pertinent medical diagnoses, whether physical or mental, that may affect the child's lessons.
10. Be committed long-term!
Often, parents want to try an instrument for a month or two to see if it works out. This really isn't enough time to decide if music is a good fit for your child! Take them to concerts, play recordings in the house, explore music game websites (there are thousands!), and put in your best efforts for practice and progress for at least 3-4 years before you make the decision to continue studies long-term.