A child who expresses interest in learning to play an instrument is ready to learn, but may not be ready for individual lessons. Very young (age 2-5) children may benefit more from music classes that stress whole body involvement, singing and sound exploration, as you find with Kindermusik programs. Some Suzuki teachers will take children that young for one-on-one lessons, but most include a class with group activities also. Weeks or even months may go by before the household witnesses the actual "playing of a song" from a child under the age of 6.
A more traditional set of indicators may be that a child is at least 6 or 7, reading words, able to pay attention to a task without constantly needing games or gimmicks to focus, and understands that there will be a daily commitment to practice. Excitement for lessons and enthusiasm to learn are also very helpful.
Mostly, you know your child. Practice is a big responsibility that both you and your child must take on willingly and whole-heartedly. Starting lessons without the desire to play enough every day to learn the assignments thoroughly, and without your inner commitment to help develop strategies to keep practice going after the initial rush has faded, will only lead to frustration and early failure. You must honestly assess your child's ability to stick to a task with long-term goals, and your ability to remain patient and encouraging while learning progresses over its inevitable bumps.
Teachers are people. Your child is a person, and so are you. The three of you will forge a very close relationship as time goes by, working and celebrating together, sometimes even crying together.
Ask your friends whose children study an instrument: do they like their teacher? Do you? Are the expectations of the teacher clear, realistic and of high achievement? What sort of music does your child learn? Are there recitals? Do you hear progress? Do you think your child sounds good? Are you satisfied with the types of things being learned? How does the teacher help solve problems in pieces, or with other issues?
When you start hearing the answers you want, and you hear a name more than once, it's time to call and enquire about lessons. Ask to observe a lesson and bring your child along. You don't have to know music to know what's important about music lessons. Trust your instincts about what you see. What are the vibes like? If it were your child up on that bench, is that the way you'd like him to be guided? Do you like the music the student is playing? Is a problem solved right there in the lesson so you can see how the teacher works? Are the assignments clear? Does the student seem happy to be there? What does your child think about what went on in the lesson?
If you like what you see, hear and feel, ask about signing on with that teacher. Most good teachers are going to be full, so be prepared to wait if you know that teacher is the one you want. Many have waiting lists, but don't just sit there and wait to be called. Check in every so often while you're waiting, let the teacher know you are truly interested and looking forward to starting.
Some teachers, like me, don't keep a waiting list. Families that start with me are the ones that have been calling every 4-6 weeks to see how things are coming along. If I know there is real interest, I'll do my best to find a them place as soon as possible.
Attend weekly lessons and take notes for you and your child to refer to during the practice week. Structure your child's day so that practicing and listening are part of the daily routine. Sit with your child during practice. Help set goals for the daily session. Delight in every step accomplished, large or small. Express your pleasure when your child plays. Do it heartily and daily.
Remember, at the end of the day, nothing means more to your child than the sense that you believe in what he's doing.