Cleaning a flute
There is no rocket science to this, the most important detail is to keep the flute as dry as possible after each use.
Use a non-abrasive slightly pourus cloth for the inside of the flute, wrapped around the cleaning rod and swab the inside of the flute to remove any moisture. Better to do it from each side of the body of the flute.
Be careful that the cloth doesn't pop out of any tone holes and snag anywhere when you pull (very rare, but it has happened).
Make sure you get well inside the headjoint and dry out the area around the lip plate. Loads of YouTube videos on this.
Cleaning the flute surface
Use a non-abrasive soft cloth, like one used for a pair of glasses. Wipe
the outside gently, and get into nooks and crannies if possible. However be careful not to snag a spring, or put any pressure on a cork that could make it come away.
When you get your flute serviced, ask you're friendly technician if he/she will dismantle the flute for you, and then take it away yourself to clean it properly before it is serviced. Use cotton buds and just a tad of Yamaha (be sure to use Yamaha, it's designed for this) flute polish and the tarnish will fly off. Also, while it is dismantled, clean down the keys and mechanism in the same manner. Then return it to your tech to do any maintenance work. If any corks or felts come off whilst cleaning, no problem, just make sure you keep them for your tech. Even if they're not used again, knowing their thickness can be useful when they're replaced.
During all this, be careful not to touch the pads, or get any polish on the pads. Also, too much polish is not good, a tiny spot goes a long way.