My 2 cents - I'm new too - got this info from two ARA top competitors:
Benchrest Rifle Cleaning Guide
Equipment and lubes
Boretech Rimfire Blend Cleaner/Lub
Remington Bore Cleaner/Polish
ProShot Grease
Proshot Patches
Nylon Bore Brush
Zippo Liter Fluid
Cleaning Procedure
1. After every 25 to 30 rounds, push four or five wet patches with Boretech Rimfire Blend Cleaner/Lub down the barrel. Let used patch fall off the end of the barrel – do not run in both directions.
2. After about 75 to 100 rounds get a patch wet with Remington Bore Cleaner and scrub just in the chamber. Short stroke it back and forth 15-20 times and then push it all the way out the end of the barrel.
3. Push wet patches of Boretech Rimfire Blend Cleaner down the barrel until they come out clean.
4. Flush the trigger with Zippo Lighter fluid.
5. Wipe off the bolt face and lugs; grease both lugs and cocking cam with Proshot Grease.
Clean the Tuner with Boretech Rimfire Blend Cleaner on a 3 inch patch
After shooting for the day, push one patch down the barrel with Boretech Rimfire Blend.
Before shooting, run a dry patch down the barrel.