Once you clean anyting with soap and water, the next step is typically a fast rinse with warm clean water to remove soap (at this point you start to get "rust", since the ph of the soap was helping to this point).
Following the clean water, I typically mechanically remove water (shake, possible wipe and blow with dry compressed air).
Next step on a complex part is to rinse in a solvent which will mix with water (alcohol works very well, either straight or part of some brake cleaners). Note trichlor does not mix very well with water nor the reverse.
Next the alcohol is mechanically removed (shake / blow dry).
Next step is to remove the alcohol (which tends to contain some moisture) with another solvent (i.e. mineral spirts / tricholor etc.).
Someplace in these steps may add a low temperature back to remove the last trace of moisture (either a small lab type oven or a space heater blowing over the cleaned items).
Blow dry and then oil all areas with general purpose oil (i.e. Break Free spray which foams up and then can be left to drip dry and possible blow out the excess).
Last step is to put on any heavier lubes at high wear points.
Well at least this is how I did it last weekend.
P.S. For muzzle loaders and other similar items, often the rinse is with warm (not boiling) water and then follow up with a water displacing oil while warm (the rust starts when the fresh water hits and boiling water on mild steel appears to make things worse) ... one of the areas where WD40 appears to work well to chase the water and then in turn is removed by the final oil (for Muzzle Loaders have had great luck with Brownells Rust Preventive #2).