Personally I disagree with the advice about the helmet, and I certainly disagree with the statement that if you fall you WILL hit your head. Most times when people come off they do not hit their heads. However, you're clearly a low-speed cyclist and it can't hurt, and may help, if you wear one.
Much more important is to avoid falling off in the first place. As others have said, you're worrying unnecessarily about derailleurs breaking or catastrophic blow-outs. They're rare. The key is confidence and practice. If I were to recommend anything specific it would be to go somewhere quiet and practice low- speed manoeuvring, because it is much more difficult to maintain your balance at very low speeds, and if you feel comfortable doing this you're much less likely to come off when having to avoid obstacles, reacting to the unexpected etc. When teaching kids to ride we used to put then on a football field and ask them to race across, with the winner being the last one to arrive. It quickly developed their bike- handling skills.
Oh, finally, don't be afraid of the front brake. The front brake is the more important, and I use it on its own most of the time. At the speeds you appear to be going you aren't going over the handlebars, and over-emphasising the rear brake is likely to cause you to skid. Use them both together, more front than back, and get used to the feel of them so you can adjust the pressure as required.
It's all about confidence. Just keep at it and in a few weeks you'll forget you were ever worried about falling off.
Last edited by chasm54; 08-08-11 at 01:20 AM.