One big thing you have to forget about is whether your kid is talented or not. Just take it off the table.
Here's the thing: for kids, it's all about the work they put into it. Yes, work. It's not all going to be fun, although it might be enjoyable. You have to help the kid set up a routine for playing every day. Or at least four days out of the week, at least ten minutes a day. If you can't do this, the chance of your kid succeeding with the instrument over the long haul drops precipitiously.
It's all about momentum. If you can build momentum from the very beginning, you will be amazed at how well your child will do. Some early success on the instrument will help her continue to stay motivated to practice, which in turn leads to more success. "She's so talented!" Uh-huh. How mysterious.
Contrast that with the alternative: Teacher and kid cover pages 10-12 in the first lesson, and kid doesn't look at them again for the rest of the week. Kid comes back, teacher realizes the kid didn't practice. Two things happen then: First, the teacher knows that practicing is going to be an issue and unconsciously holds back her best stuff. Second, the kid gets assigned pages 10-12 again. There goes the momentum. It takes a lot to fix this problem: The kid has learned that practicing is not a part of taking music lessons.
I cannot overstate the importance of the crucial first few weeks of music lessons. Developing the practice routine is a lot like practice itself: getting it right on the first try is exponentially better than fixing mistakes. It takes seven repetitions to learn a given musical figure, and thirty-five repetitions to un-learn it once it's been learned incorrectly. Likewise, a child who gets off on the wrong foot has learned incorrectly that learning an instrument is about attending the music lesson and nothing more, and once a child makes this erroneous assumption it is extremely difficult to get him to a place where he is learning music for real.
This is such an avoidable problem. But how involved should you be in preventing it? This depends on the age and temperament of your child. For most children under twelve, you can pretty much assume that you will be overseeing daily practice, reminding your kid to do it, and providing rewards for doing it/consequences for not doing it. If you are not willing to be this involved, it is unlikely your child will succeed at music lessons, even if it was his idea.
For children twelve and up, it is a judgment call. You have to choose your battles on everything at this age, so maybe you want music lessons to be something you don't have to be involved in or fight over. If this is the case, you have to decide whether you are okay with spending the money on lessons even if your kid isn't practicing. The exception to the momentum rule is that sometimes when teens are given the space to develop their passion for music, it will bloom on its own.
Be prepared for the teacher to have a say in this, too - your teacher might not be willing to work with a student who is not practicing. Children who don't practice are not fun to work with.
Many a parent wants his kid to be a natural - our culture fetishizes prodigies and overnight successes. But practicing music is really about training the student to feel pride in a sense of accomplishment - to derive pleasure from a sense of progress after a period of hard work. That's not glamorous at all! So what's the shortcut?
You guessed it: momentum. The funny thing is, momentum starts to look eerily like talent when developed early and cultivated faithfully.