Suzuki is fun for kids. It's a Japanese music learning system. They simplify the welknown classical musical pieces into beginner to advance levels for children to learn to play.
It would be more fun than the ordinary learning because otherwise the children would only play some boring short study pieces instead of music pieces that are familiarized by everyone.
Suzuki lessons can be started very early, and as early as about 4 years old, and very often, children learn in a group so they can play the music together with friends and have more fun. You can also buy the Japanese Suzuki violin for your child. They have very very small size 1/16 of a full size violin, and the quality is better than the Chinese made ones (sorry, I know there are good Chinese made ones, but most of the ones buit for children produce terrible sounds)
Violin is not my favorite instrument, but I think it's fun for kids since they can carry it around to play, and the cost for the equipment is comparably cheaper than other bigger instrument. Indeed, a piano's sound is much better to listen to than a violin for parents.
Indeed, parents would suffer a lot if your children are learning violin, since the sound is really unbearable most of the time when the kid is playing, and your neighbour will probably hate you for this.
You can buy a "mute" to put above the bridge of the violin, and that will lower the volume down a lot.
A 6 years old child usually use the 1/4 size, but it depends on the height of the child.
A 9 years old could fit to 3/4 size
A 12 years old probably can use the full size.
I don't think the non-full sized instrument would be a good investment, so you can just rent one from the school and get the right size as your child grows. Children do damage their instruments as they are growing up.
When I was a kid, my school let us practice 2 hours every day, but most kids just use the time to "play", and I think piano would need more time to practice since the music note for violin is a lot more simplier.