I start by pushing a patch or two through using something like Hoppe's #9 or Butch's Bore Shine or Shooter's Choice or Break-Free CLP or FP-10. Then after I've gotten most of the rough crud out I'll push a brush through from the back end a couple of times using the same cleaning solution. I usually spray the brush clean with Gunscrubber or some such thing between passes and I don't ever dip the brush in the bottle if it's dirty - maybe pour a little in the lid if I want to dip it, but then I can rinse that out.
If the brush gets too worn you can wrap a small patch around it, wet it with whatever you're using and then push it through from the chamber end. Don't force it because it might get stuck and if you're forcing it the rod could/will bend and rub against the inside of the barrel. Wipe the rod with a clean patch to keep it clean. Dirty rods and aluminum rods act like sandpaper inside the barrel and will eventually do damage - especially at the muzzle.
Don't let the brush or patch pop out of the muzzle and drag the rod on the metal as you keep pushing for 6 or 8 inches. You're cleaning a gun, not churning butter (I think my age is showing.) The muzzle end needs to be protected from abuse because it's the last thing the bullet touches on its way to the target. Picky people will unscrew the brush and then pull the rod back through very gently. Why drag all that dirt back into the barrel and scratch up the crown.
Some people never clean their rimfires or maybe only when their accuracy falls off. Less is better than too much. <= Read this again.
If accuracy falls off you can always go home and scrub it later.
If you take the bolt out, and if it's dirty, you can take a clean toothbrush and scrub the bolt face gently with your cleaner and then hit it with the spray cleaner to get the mess off. Wipe out the inside of the receiver with a patch.
Take a patch or a small soft rag and gently wipe everything down with gun oil inside and out (it's humid in Virginia so I go a little heavy on the oil.) Don't forget to put a little in the barrel - Gunscrubber takes ALL of the oil off the metal. A thin coat will do and do not shoot a gun with a barrel loaded down oil. If in doubt push a dry patch through before you leave home to go shoot it.
I don't know what the other thread said, I've been drinking coffee since 6 a.m. and felt like beating on the keyboard.
Guns really aren't all that delicate, but you might as well baby them a little because you never know when you'll end up owning one for 30 or 40 or more years.
John