I bought some special oil at my local music store for oiling the keys. I don't need to oil the keys of my new flute yet, but I should I think oil the keys of my old flute (well, it's not that old, just 2 years and a couple of months old). (I should have done this before, I know... )
The BEST method is to let a repair technician do it. The reason is that to do a really proper job of oiling requires the flute to be dissasembled so that all of the steel shafts and rods can be cleaned. After cleaning, it's ready to receive a good coating of fresh oil. This is very similar to changing the oil in a car engine. Over time, the oil will start to get dirty and the particles in the oil can cause wear. A flute is not exposed to the same harse environment as a car engine, but it's still a good idea to have that service performed annually. Some repair techs call this a Clean, Oil, and Adjust (COA). Some shops call it a play condition (PC).
If you insist upon oiling it yourself, you can apply little oil at the joints between each key shaft and between the key shafts and posts. Put a very small drop of oil on a needle or toothpick and touch the point to a joint to apply the oil. The oil should wick into the joint. Repeat for each key joint. The most important thing is to be sure that the oil does not run onto any of the pads. Keep a tissue handy to immediately blot any excess oil. If any oil runs under the keys, blot it with the tissue or use a cotton swab if you can't reach it otherwise. Be careful that you don't catch the tissue or swab on a spring and cause it to become unhooked from the spring cradle.
If you don't feel comfortable with the above process, take it to a flute repair technician.
"Never give a flute player a screwdriver."
--anonymous--