1) Two Bucket Method: When the bike is extremely dirty and needs to be properly cleaned
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http://www.autogeek.net/exterior.html)
This is how I wash my bike and car. The two bucket method helps to keep the amount of dirt on your wash mitt low so that you're not scratching the paint.
First I wet the bike with the garden hose, then I use a wash mitt to wash the bike. I keep the bike wet with the hose until I'm ready to dry it.
I use Meguiar's Gold Class car wash because it doesn't have any wax. I find wax turns my textured black plastic pieces gray because the wax builds up the in the recesses of the texture.
http://www.canadiantire.ca/en/pdp/meguiar-39-s-gold-class-car-wash-0394513p.html
For the wash mitt, I use something with a long nap (long fibers). This also helps to keep any dirt that's been picked up by the mitt away from the paint of my vehicle. I bought mine from Canadian Tire but I couldn't see anything that looks like what I have from just looking at the photos. This mitt does look interesting though:
http://www.canadiantire.ca/en/pdp/simoniz-platinum-chenille-wash-pad-0397065p.html
For drying, I use microfibre towels. Something like these:
http://www.canadiantire.ca/en/pdp/simoniz-microfibre-cloths-8-pack-0396620p.html
For the chain area - meaning anything that get's really nasty from chain lube - I use kerosene. I have one of the Tirox 360 degree brushes which I think works really well on the chain. You can get that at GP or Riders Choice.
I have a couple of different brushes that I use to get into the nooks and crannies. I'll either use soap and water with them or kerosene, depending on what I'm trying to clean up. Kerosene is really great against the oil and lube grime.
I also wash my tires occasionally. For that I use a brush and Spray Nine. I find that Spray Nine rinses away completely and doesn't leave any residue, but it does remove any lube that may make it onto the tires. I don't over-lube, but eventually grime builds up on everything.