I agree with rcmodel.
Clean everything relatively well, then leave it alone. I'd avoid the brushes in the bore, too. Cleaning can contribute as much or more to barrel wear than shooting.
Bore snakes are great, I use them on all my rifles when at the range.
At home, I push a patch dripping with solvent down the bore, and let that soak while I tidy up the rest of the rifle. Make sure to clean your boltface, lots of people ignore that and wind up with extraction problems. After 10 minutes or so, swab out the solvent. If patches are still coming out grungy and there's no solvent left in the bore, I run another wet patch down.
I stop as soon as patches are coming out fairly clean and there are no visible spots in the bore.
If it's a .22 rimfire, it gets cleaned once a year, boresnaked after each range outing. Their barrels are usually made of relatively soft steel, cleaning can and does kill more .22 barrels than shooting ever will.