First of all, don't worry about cleaning. There is nothing in modern ammunition that will leave residue that is harmful to a firearm. The exception is with cast or swaged lead bullets which may cause some bore leading.
As for cleaning, clean the bore and chambers, wipe down the exterior and lightly oil. Do discard patches or oily rags, or else store them in a steel container to prevent spontaneous combustion.
As to solvents, I use Hoppe's No. 9, and have done so for over sixty years, see no reason to change now. As a final touch, I oild the gun rather heavily, then wipe it down with those blue paper shop towels from AutoZone or Home Depot.
The oil I use is a home brew of 50/50 (more or less) of automotive motor oil an Three-In-One machine oil. This has worked for me for many years, and hopefully, many years to come.