Cleaning a saxophone will not only help it produce a clear tone but will prolong its life as well. A straight saxophone, such as the soprano, requires special tools to be properly cleaned. However, most other bell shaped saxophones can be cleaned quite easily using a standard cleaning kit. The information here addresses the more common bell shaped sax.
Clean the mouthpiece by removing the reed and ligature and running cold water through it. Then pull a clean lint free cloth through the mouthpiece to dry it and remove any particles the water missed.
Swab the neck by inserting the brush attachment from the cleaning kit into the larger opening on the neck. After brushing out the inside to remove any foreign particles, run the rag ball on the opposite end of the brush through the neck to swab it.
Find an attachment in your kit that looks like a brush and cloth on a long string weighted on the opposite end. Put the weighted end into the bell of the saxophone, and turn the sax upside down bringing the weighted end completely through the body and coming at the narrow end where the neck is attached. Carefully pull the swab through the body and repeat this process several times.
Clean the valve pads by sliding a piece of paper or pad cleaner underneath the pad. Then close the valve and slowly pull the paper out. If a valve pad is torn, don't try to replace it yourself. Take it to your local repair shop and have a professional take care of it for you.
Grease the corks with a liberal coating of cork grease and massage the grease into the cork to condition it. Doing this regularly is necessary to help maintain a good seal. Be careful not to do it too much and saturate the cork which will cause it to deteriorate.
Reassemble your saxophone, and it will not only smell good but will play beautifully.