I agree, Avguste, we pianists MUST set a high standard for ourselves. It's habit for the public and (ironically) other business owners to see us as good-deed-doers and not the entre-pre-neurs we actually are.
Here's a personal story.
My first 'gig' was every Friday and Saturday night at a nice Italian restaurant (I got free food, too; definitely a plus!) playing mostly classical, traditional Italian songs, a bit of Jazz/improv. I was paid whatever tips I got plus a predetermined amount for a predetermined time and that was that, so I thought. A couple of months in, the owner (badabing!

) called me into his office. He wanted to bring in a violinist to play along with me. It sounded like a good idea, it
was a good idea at the time. So the violinist and I meet; she's an attractive, talented violin instructor. After playing that Friday, we walk into the restauranteur's office, expecting the predetermined rate. Instead, I'm treated to this explanation of why my payment is being cut almost in half. Blah, blah, badabing. Blah, blah, opportunity; I did not want to hear it. I was in a very tough spot; this gig was paying very well and it would still pay well, even if I accepted the pay cut. I told the owner and the clueless violinist I would sleep on it. The next day I tried to reason with the man, that splitting tips that were not increased by the violinist was not fair to me, that splitting our payment was not fair to either of us. He had made up his mind. Unfortunately, I had made up mine, too. I bid him farewell and left him my card, in case he would have a change of heart. He never called me. Lowering my charge on such a whim, would have damaged my current reputation as a first-class hardass.
A month later, I got a new gig at a different restaurant. At age 21, I made a strong effort to appear as professional as possible in my dress-code and demeanor. My suggested rates were clearly stated and I left the owner of the new restaurant with a video of me playing elsewhere. He liked what he saw because I succeded in getting him to agree to a one year, 100 night contract. A few months later, he asked me to play Sundays at brunch, at 2/3 regular pay. After some discussion, I agreed. Behave as a professional and you will likely be treated as one.
Whatever you end up charging, PUT IT IN WRITING!!! A simple form will do for one time events, like weddings. A formal contract is best for regular gigs and concert engagements. Business owners usually won't agree to a long-term contract, that's OK. Try to agree on a short-term contract. Just make sure you come to a meeting of the minds (and put that mind-meeting on paper) before you provide any services. If you fail to do that, you will have to endure the odd surprise. I don't like surprises, do you?
The same goes for giving lessons, although you are far less likely to get an argument from a parent than from Vito Corleone, you must put the agreement on paper. (In Louisiana, the cost of living is very low, I charge thirty dollars an hour for lessons. In San Francisco, I imagine the amount would be higher.)
...and to all a good night!
Pete