I wash and wax my bikes regularly. I use dish detergent most of the time, but for really tough areas Simple Green works well. A coat of car wax does a good job of making it easier to clean the next time. It also keeps the bike looking nice.
This is part of a sequence of pictures I used in a bike maintenance class. Gentle, fine spray from a hose will not hurt anything.
A toothbrush, a small brush, and a sponge are all you need, unless you are doing a deep cleaning and lubrication.

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