Both the inside and the outside of the flute need to be cleaned. It is important to clean the inside because you don't want your metal to corrode. Cleaning The outside is important because you don't want the flute to look gross or the silver plating (if you have that) to wear down.
Your flute should come with a rod that has a long skinny hole at the end of
it. Go to a store and buy a handkerchief; put about half
of the handkerchief through the hole in the rod and put the rod and the
cloth through the flute a few times. When you do this make sure that the
flute is in the three separate pieces; clean all three pieces. The head joint
will be the hardest of the three because the rod will not go all the way
through the entire piece.
It really depends on how dirty your flute is. If you just purchased a used flute, or you have never cleaned the outside of your flute and it is really dirty, you might consider having a professional clean it. If it is shinny but just a little dirty, you can use a silver polishing cloth to clean it; you can get these at almost any music store. You can also use that same handkerchief that you use for the inside on the outside. Saliva is a good cleaner for the outside of the flute!
Under no circumstance should you ever run the body of your flute under water. The pads on your flute are not meant to get wet, and if they do, they will expand and could cause damage.