It includes my coaching rates, ice charges, admonition to practice, schedule vagaries, cancellation policy, lesson rules (like late arrival), and a statement about changing coaches. It has a liability waiver, for what it's worth. It has contact information for both parties, and the skater's birthday.
I get a copy signed by the parent or adult student at the first lesson. They get a copy to keep.
A second page talks about decorum, parent do's and don'ts, proper attire, competition, and other FAQs.
Such a contract is highly recommended by the PSA, and yet I know for sure of only 3 coaches who do this. I am the ONLY non-competitive coach I know who does this. When I first starting teaching, I asked some more experienced coaches if I could take a look at their contracts so I would have a model. Most of them didn't use a contract; several actively disdained the idea (and used it to add to the myth that I didn't know what I was doing). The ones who had one refused to share it, because it was "proprietary." (At which point I simply went to the parents and asked them if I could take a look. And they showed me. Really, coaches can be such idiots.) I'm guessing that some of the coaches who told me they don't do contracts actually do, but didn't want anyone to know, for some reason that I can't fathom.
All elite coaches (those with Nationals-bound students, generally) have contracts with their high level students. Contract disputes have been behind some famous coach-skater break ups. It's been speculated that a contract dispute is what split Michelle Kwan from Frank Carroll, and possibly cost her the Olympic gold.
Of course, I'm not trying to get a percentage of your future earnings, as I'm pretty sure there won't be any, at least not from skating (not if I'm your coach, anyway, lol). I'm just trying to make it clear to everyone what the deal is.
The coach seems like your friend. She loves your child. She's a gas to talk to.
But this is a professional relationship. If your coach doesn't have a contract, at least ask her for her rules in writing. If she won't do it in writing, sit down with her, and ask about them and you write it down: fee structure and billing practices, arrival policy, coaching change policy, missed lesson policy, make-up lesson policy, skating attire policy, competition policy, music policy. Believe me, every coach has a policy for these things, whether they know it or not.
If they tell you "don't worry about it," insist. If they refuse to let you know how they deal with these things, find another coach. (This is not the same as a coach who hasn't thought about it-- many young coaches won't have realized how important this is. That means you can be the one to help a young coach develop her policies.)
Does your coach have a written contract? How did you feel about it? What are some good points in the contract?