Good for you on the helmet. It does not matter how far you are from home--you can fall off your bike anywhere. I don't understand why some folks (not you, just in general) feel that they don't need a helmet because they are just going on a short ride. It's like thinking that you only need to where a seatbelt when you go for long drives. When I had my bike accident and had to be airlifted to the trauma center, I was only a mile or so into my ride.

Your wrist might be aggravated by a couple of things:

1. putting all your weight on your arms and hands. This is common mistake that beginners (and others) make. You'll want to do some core strengthening exercises so that you can support your body with your core and not only with your hands and arms. In the meantime, make sure that your elbows aren't locked. They should be slightly bent to absorb the bumps.

2. your bike's top tube might be too long. Do you feel like you are too stretched out? You can do some adjustments to the stem (new shorter stem) and to the seat to compensate a bit.

Have fun riding!

EDIT: celery musta posted the same time as me. she already hit all the points.