While I'm not a musician myself, we do have one son who played Bassoon in symphony, upright bass and bass guitar in jazz, and percussion in the marching band. What that equated to was an uncountable amount of music lessons. Though that what I learned was the following as a parent:
1. Just like athletics, any musician needs to start developing their musical "core" That means learning to read music well, music theory, and recognizing tone. I know that our local HS requires those in percussion to take at least 2 years of piano before being allowed into the percussion group for that very reason. I'd suggest starting on the piano rather than the violin if that's possible.
2. What I personally think of a teacher other than as it relates to safety isn't nearly as important as how my child "clicks" with the teacher. How I learn isn't necessarily how my child learns, what inspires me isn't necessarily what inspires my child. Try to match your child's learning style with the teacher. Does your child respond better to a nurturer or someone that is more strict? Someone who is goal driven and takes a direct path or someone with more flexibility in their approach?
Youngest son had a old school piano teacher, a very young flexible in his approach jazz musician as a percussion instructor, and an old hippy HS symphony instructor teaching him upright bass and bass guitar. He got along with all of them wonderfully and appreciate the diversity among them. Find the right fit is the most important thing.
3. Your child WILL eventually resist practice, help them to work though it. They'll learn a great deal about how consistent practice over time equates to excellence. If nothing else learned though learning an instrument, that was the key take away for my son and a very valuable one at that.
4. The music community is much more welcoming of ages differences than the athletic community generally speaking and they nurture each other a great deal. Playing an instrument at school opens a lot of friendship doors.
Enjoy!