Taking Care of Your Flute
The main things you need for good flute care are the tuning/cleaning rod that came with your flute and two thin micro fiber cloths.
Cleaning the Inside
After every practice, thread a corner of one of your cloths through the slot in the cleaning rod. Wrap the cloth once over the end, and loosely wrap the remainder of the cloth around the rod. Create enough bulk that the cloth will wipe the inside of the flute without being forced. Disassemble the flute. Next, hold the end of section with no keys. Carefully push the cloth through each section a few times to wipe away moisture. Take care not to bang the rod on the end of the inside of the head joint.
Wipe the Outside
Next, while holding each section by the end, not by the keys, wipe the outside of the flute with the other cloth. First, close each key by pressing on it with the cloth and wiping the closed key with the cloth. Wiping each key while closed will avoid pushing the keys out of alignment. After wiping each key, gently wipe the key arms and rods. Finally, wipe of the smooth parts of the body and place each section in the case.
About once a year, you may use a good treated cloth made for your instrument to remove tarnish. Such cloths are treated to remove tarnish without attacking the metal. Treated cloths are also easy to use without contaminating the action or pads. A good tarnish removing cloth for flutes is made by Blitz. The Blitz cloth includes one side to pull tarnish away from the metal and another to wipe off the tarnish residue. Avoid using silver polish, rubbing alcohol or other home concoctions as they can impact the metal and are too difficult to keep away from the pads and action.
Polishing may not be necessary if you are good about wiping off your flute everyday. Also, you may want to leave polishing to a technician if you have an expensive professional instrument.
Steer Clear of Pad Savers
Do not use pad savers, those things that look like giant pipe cleaners that are left in the flute while in the case. If you use them to dry the inside of the flute and leave them inside the flute in the case, it has the same effect as not cleaning your flute. If you dry the inside of your flute with a cloth and insert the fluffy pad saver, it serves no purpose.
Use oil sparingly if at all. As any good flute technician (or gunsmith) will tell you, oil applied to metal does not go away. Oil builds up and attracts dust. Eventually it breaks down becoming sticky and gummy. The very thin coat of oil applied to your flute action by the maintenance technician keeps everything moving smoothly. See a technician if you find your instrument needing oil to keep keys working.
Sticking keys are often the result of not getting the moisture out of your flute when you clean. If you are cleaning regularly and still have sticky keys, try a different cloth. Your cloth may not be absorbing moisture. When a sticky key persists, place a very thin paper under the pad and close the key several times. Cigarette paper or tissue works well. Be careful not to pull on the paper when the key is closed. You may tear the pad.
Tuning is checked and adjusted by inserting the tuning/cleaning rod, notched end first into the head joint until it gently touches the end. If your flute is more or less in tune, the notch will appear at the center of the embouchure hole. Adjust the position of the notch by tightening the end cap to move the notch left or loosening and then pushing on the end cap to move the notch right. If the end cap will not tighten or push in, take it to a technician. Use a tuning app on your phone to check for sure. A good app is Soundcorset. Soundcorset also has a metronome.