The most important cleaning you should do on your new Savage, is the one you do BEFORE you shoot it the first time. Get the dirt out of it. Most of mine have arrived surprisingly dirty from the factory.
After that, pick a method out of a hat and go with it.
For me, I will never waste another day doing the shoot-n-clean thing fifty times on a factory Savage barrel though. Just shoot it and clean when accuracy becomes unsatisfactory. You're not dealing with a benchrest gun here. Totally different animal.