Is there a good way to clean my saxophone without buying sprays or Windex, but without damaging my instrument, and using common household objects?
To clean the inside you need a pull-through, which is a cloth attached to a string with a weight on the other end. See your local music shop or online. Clean the mouthpiece especially carefully. Wipe the reed and put it out to dry before popping it back in the reed keeper.
To clean the outside use a soft cloth. If it's really dirty (don't let it get that bad!) use metal polish. You can get polishing cloths already soaked in polish, even gloves. Try not to get it caught in the moving parts; you can dislodge the little springs if you are not careful.
If it's really disgusting take it to your local sax-doctor and have it overhauled.
It is important to keep your instrument clean, especially the mouthpiece. Always clean after use, and leave to dry out. Do not keep it in the closed case.
Having said that, most jazz musicians like to keep the outside of their instrument looking scruffy with a certain patina. Do not be fooled; it is clean on the inside and the keys all work properly.