Instructables
Picture of How to Clean a Trumpet
The following is a set of instructions on how to safely and efficiently clean a trumpet.  The methods described are those I have found to yield the best results.
 
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Step 1: Before You Begin

Picture of Before You Begin
Before you begin cleaning your trumpet you will need to gather and take stock of your supplies.  You will need a "wire-snake" brush, a mouthpiece brush, petroleum jelly, valve oil, a lacquer polishing cloth or a silver polishing cloth (depending on your trumpet's finish), several towels, and non-bleach dish soap (Bleach can be corrosive to the instrument's finish.).  These supplies are displayed and tagged above.  You will also need to decide on a wash basin, such as a bath-tub (recommended) or large sink.  The wash basin must have running, heated water and room for maneuvering the trumpet safely.  Be prepared to apply only lukewarm water, however, and not water that is too hot.  Very hot water could cause the trumpet's finish to flake.  You will be reminded of this every step.  Finally, acquaint yourself with the following terms if they're unfamiliar:

Bore - the interior chambers of the instrument that define air flow path
Lead pipe - the pipe connecting the mouthpiece and tuning slide; part of the horn
Bell - the final, conical section of the trumpet; part of the horn
Valve casings - the chambers that contain the valve pistons; part of the horn

Other important trumpet parts are depicted in the next step.

Step 2: Disassembling the Trumpet

Picture of Disassembling the Trumpet
Disassemble your trumpet and place the parts on a towel.  The towel will prevent the parts from getting scratched and provide a clean surface.  Make sure to locate the towel in a safe place where the trumpet won't get stepped on or otherwise damaged.  The disassembled parts, tagged above, are the mouthpiece, the 1st valve piston, the 2nd valve piston, the 3rd valve piston, the 1st valve slide, the 2nd valve slide, the 3rd valve slide, the tuning slide, three valve caps, and the horn.  The trumpet may have several screws that hold the 3rd valve slide in place and a "grime gutter."  If so, set these aside in a safe place.