Well, the standard powder solvent for YEARS has been Hoppes #9. But there are lot of others- Ed's Red for one. You might also want to try one of the spray cleaners, like Gunscrubber (keep it off finishes on wood) Remoil is a decent light lube, as is Break Free. Visit your local sporting goods store, see what they have.
Advice? Make sure it is unloaded. Then check to be sure it is unloaded. Did I mention to be sure it is unloaded?
Only take your rifle down to the point that it NEEDS to be taken down. Many a gunsmith ahs been presented with a bag of peices parts, and a request to put it back together.
Where possible, clean the barrel from the rear (breech) and not the front (muzzle) to reduce damage/wear to rifling at muzzle. Not possible with all rifles, so take it easy. Run brush wet with solvent ALL the way thru, then ALL the way back. Repeat about 10 times, let set for a few minutes, run patch thru, replace, repeat until clean. Be sure to clean chamber. Old toothbrush and Q tips are handy for getting into nooks.
Do not over lube. Avoid scrubbing exterior with anything other than light oil and a cloth. I use wax (Kiwi neutral shoe polish) on wood. Do not let lubricating oil soak into the wood