Currently, I have 4 children that are learning to play an instrument using the Suzuki method (2 - violin, 1 - cello, 1 - piano). I have been working with at least one of my children for 5+ years now and have learned a few tips along the way. I hope this helps you in some way.
1. Listen, listen, listen and sing. There really is no substitute for having your child listen to the pieces played properly. They progress much faster when they can hear the song in their head as they play and/or can sing it as they play. I find it helpful to be listening to the piece that they are currently learning and the piece that is on deck.
2. Build your practice time into your schedule. After fumbling around for a few years trying to work in practice when it was convenient, I have found that having a set time every day makes it easier on me and my children to get consistent practices. We all know when it is time to practice and I no longer have to twist arms or bribe my children to practice.
3. You must be with them when they practice. I have made the mistake of sending my child off to practice by themselves. I find that when this happens and there is no accountability that my children do the following: practice what they like (often whatever they are already good at), give up at the hard parts, spend more time and get less practice, develop bad habits, view practice as a bore / punishment, and once your child spills the beans at the private lesson…. you are busted! My children have grown to enjoy their undivided attention that they get during this time.
4. Try to read your child's cue's and don't force the issue. I have learned that Rome was not built in a day. "Just one more time" could actually do more damage than good. If your child has had enough then call it a day. It is better to do one less repitition today and have an eager learner tomorrow than it is to force a disinterested or defiant child to complete one more repetition today and get a begrudged child tomorrow. Learning an instrument takes time so don't put to much pressure on yourself.
5. Don't try to be the expert. Simply put….because you aren't. Don't feel the need to be always right. If you don't know, say so….if you are wrong, admit it. I know I am not a music expert and even if I were, the odds that I am expert at Violin, Cello and Piano would be slim. Figure it out together and if you are stuck, make a note and be sure to ask your question at the next private lesson.
6. Equipment Tips. If the instrument sounds out of tune, it probably is. Stop and tune it so your child is not learning to hear the incorrect notes. Also, if you have an instrument that has a case, figure out a way to safely store the instrument that does not include putting it back in the case. This makes the beginning and ending of your practice time quicker and more efficient and also allows for impromptu playing during the day if there are only a few minutes to spare.
7. Get your child in a group. Most cities should have Suzuki groups available for your child. This is a chance for your child to play their instrument with other kids at varying skill levels. You child gets to feel apart of a team, learn from others, share their accomplishments, and maybe most importantly, make friends with other kids who play their same instrument and realize they are not "the only one" and it is "cool" to play an instrument.
8. Have your own concerts. Nothing is more motivating to my children than playing a concert for Grandma and Grandpa or their cousins. Allowing them to showcase their new talent in a loving environment in front of people that they love and want to please is very rewarding.
9. Find videos online to share with your child. It was very motivating for my kids to, from time to time, show them a video on youtube of someone playing their instrument. I vividly remember watching two guys play "Smooth Criminal" with their cellos and watching a girl who plays the violin and dances at the same time. This opens up their mind to the possibilities of what lies ahead and it creates the same response as a kid watching a basketball game and wanting to go outside and shoot hoops.
10. Try to make this about more than your child. I discovered early on that being a Suzuki parent was going to require some time and dedication. Why not view this as a learning opportunity for yourself, that you get to share with you child, instead of a sacrifice and investment in your child's future? I decided that I was going to learn the piano. Now I view music practices and lessons with my kids as my own music education and who knows, maybe I will be able to play the piano someday, without ever paying for my own lessons!
I hope that one or more of these will help you as you work with you child. There is no 100% correct way, so if any of my suggestions don't resonate with you then disregard them. Take them with a grain of salt and figure out what works best for you, your child and your family.