Try to touch the piano keys lightly and gently. If your fingers are rigid, stiff, with the anxiety of trying to get the right notes at the right time, it will be much more difficult to get it right. A relaxed hand is much easier to get the the right place at the right time.
Have a specific plan when you practice. Know where you're headed and know what it is you want to be able to learn in each practice session. If you don't follow a plan it will be very easy to jump from one place to another and that will cause you counltess of wasted hours.
Vary the work to avoid boredom, having other, perhaps easier, pieces to hand
Don't worry about making mistakes. If all we did was worry about making mistakes, we would never get anywhere. We would live in constant fear!
When you are learning a new piece of music for example. If you were to know in advance that you where going to make 25 mistakes before you could play the piece perfectly, would you worry about making these mistakes?... NO! You would want to make these 25 mistakes as quickly as possible... wouldn't you? What I am trying to say is each mistake you make brings you closer to your goal.
As long as your ear can recognize the mistakes, you can correct them and move on.
What are the teacher's musical degrees? Degrees don't guarantee a good piano teacher, but they do suggest a teacher who has been exposed to a wider range of musical knowledge. Keep in mind that there are a number of types of piano-related music degrees (piano performance, piano pedagogy, general music, etc) which have different emphases and give different perspectives to the teacher. Some teachers will even have doctoral degrees in music (D.M.A. - Doctor of Musical Arts).
What kinds of music does the teacher give instruction upon? If all you want your child to do is plunk out a few tunes, you probably don't want to have a teacher who emphasizes classical training, even though the classics are still probably the best vehicle for really learning the piano.