Based upon what I've gleaned from the World Wide Web, I've settled on a method that works for me, and that I have confidence in. When I receive a new rifle, before I shoot it or do anything else, I break out the JB Non-embedding Bore Paste, slather some on a brass brush or cotton swab, and go down and back in the barrel like twenty times. I'll clean this out with Hoppes #9, and final clean with patches. I plagiarized this method from a blog whose author I respect greatly, and from others who use the same method. One gentleman whose opinion I also trust suggests that using the brush with the paste is a waste of effort and suggests using a swab, and I've done that also. I use JB paste because it's designed for the job, but I'm sure not gonna dispute those who use other materials. Again, if a guy uses a particular product or method and has excellent results, and has confidence in it, well, who am I to say?
I will strongly aver that the initial cleaning is just one of the most important things you can do. Even the highest quality air rifles seem to come from the factory with a lotta smutz in the barrel, so I don't want to start laying lead in that barrel before I get the poop out of it! To me, that's just common sense! On the other hand, I've had several come spankin' clean from the factory...