Cleaning a saxophone is a relatively simple process, provided that your saxophone is the standard, half-bell shape. Straight saxophones, such as a common soprano saxophone type, require tools not mentioned here. This article will assist you in understanding the mechanics and basics of cleaning your saxophone to maintain a clear, deep tone.

[edit] Steps

Clean your mouthpiece. Remove the reed and ligature, and use the mouthpiece brush to remove any foreign material inside the mouthpiece. Run lukewarm water through the mouthpiece at the sink. Finally, using a clean, lint-free cloth, pull it through the mouthpiece to dry and remove any particles missed by the brush. This may take a few passes.
Swab the neck. Use the brush attachment (looks like a flexible metal hose with a rag ball on one end, and a small brush on the other) and pass it through the larger, base opening on the neck, coming out on the narrow side to which the cork is attached. Brush out the inside to remove any foreign particles or bacterial growth, and use the swab last. You can run water through the neck as well, just be sure that no water comes in contact with the cork, or it will swell and deform. Be wary of the pad on the octave valve as well.
Swab the body. With a standard cleaning kit, there is an attachment 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 to which the neck is attached. Gently pull the swab through the body, repeating this process several times. If possible, hold the keys down as you do this. It is not unusual to see a slight green color on the pad after a few passes. This is corrosion on the inside of the saxophone, as brass rusts green due to the copper.
Check & clean the valve pads. There are many on the saxophone, so this may be the most time consuming step. Visually inspect each pad, looking for wear or tears. If a pad is torn, take it to your local repair shop for replacement. Using a piece of paper, a dollar bill, or a pad cleaner, slide the paper underneath the pad, close the valve, and slowly pull the paper out. This helps remove foreign material from the pad.
Tighten loose screws. Most screws on a saxophone are of the flat head, not phillip's. You can tighten loose rod screws safely, but do not over-torque them. If this happens, you may not be able to press keys such as those necessary for a high-D or F#.
Swab and grease the corks. Dry the cork on the neck completely, and add a liberal amount of cork grease. Rub the grease into the cork to 'condition' it, and add another light coating. Do this weekly, and you'll maintain an impressive seal. After some time of doing this, the cork will get saturated with grease; do not grease further, or you will hasten the deterioration of the cork. Don't try to grease the little bits of cork on the ends of actions; they're there for padding.
Clean swab every month for removal of wastes and for decreased build up of corrosion.
Reassemble your saxophone. It should look, smell, and play beautifully!

[edit] Tips

Purchase a "Sax Saver" from a local store. A sax saver is a body insert that looks like a furry rod that is placed into the body of the saxophone during transport and storage. Not only will the saver help absorb excess moisture, you won't have to worry about losing another neck cap. However, the use of these are VERY controversial, as many claim that they absorb the excess moisture and let mold grow in them, as well as in the sax. The increased moisture may also promote rusting. Make your own judgement.
Clean your saxophone at least every other week. A good rule of thumb is to clean your sax whenever you need to replace a reed.
At least swab your sax every time you play! Don't put it away wet; that will promote mold and maybe rust.

[edit] Warnings

Do not attempt to oil, remove dents, replace pads, or use scratch-removers on the finish of your saxophone. Leave these things to a professional. If you have a rental instrument, in most cases these services are performed free of charge.