Revolvers aren't as common for carry pistols as they used to be, but there are still a lot around. And that's not counting those used for hunting and targets. They're reliable, versatile, and generally handy. Cleaning them is different from cleaning a semi-auto.
For this I'm using a Smith & Wesson .38 Special.
First thing, open the cylinder and make sure there's nothing hiding in any chamber.
For the barrel, wet a patch with your favorite CLP or whatever, and push it through the bore.
Let it soak while you attend to the cylinder. Use a bigger patch, because the chambers are larger in diameter than the bullet. Hold the pistol so you're supporting it, pick a chamber, and push the patch through that chamber.
It'll be pretty tight right at the front, because there the chamber narrows from cartridge diameter to bullet diameter.
Remove the rod, take the patch and put it back on the jag, and do the next chamber. If it's really dirty, and the patch looks nasty, use a fresh one as needed.
Once all chambers have been patched, take a cloth, wet with your cleaner, and wipe the front and back of the cylinder.
Now take a cloth, and clean out the inside of the frame.
You might need to use a craft stick or something to push it tight around the barrel,
otherwise a finger works.
Back to the barrel, push a dry patch through, then take a look. Helps to hold the piece so light reflects off the back of the action, or use a flashlight. Unless you had some really dirty ammo, or it's had a LOT fired through it, chances are that's all the cleaning it'll need.
Go to the cylinder, push a dry patch through each chamber, and inspect. If any that need more attention, you can hit them again, or use a brass or bronze brush to help get any stubborn fouling out. If you have to do this, it helps to have a rod with no swivel action, as you can rotate the brush to better scrub the chambers out. Can even go to a little larger diameter brush if need be.
Then wipe off the front and back of the cylinder.
Now dry cloth, and wipe out the inside of the frame.
Wipe down the outside of the barrel and frame, the ejector rod, anywhere that looks like it needs it.
A lot of people do prefer to remove the cylinder for cleaning. On the S&W, that's simple. See that front screw on the sideplate on the right?
Open the cylinder, and line one of the flutes on it up with this spot on the frame(shown with the cylinder removed).
On models that have a ejector rod that's the same diameter all the way, you can slide the crane forward and off the ejector rod. Older models often have a larger-diameter head on the rod, which means you'd have to remove the ejector rod to take the crane completely off. We're not doing that today.
More advanced: say you picked up an old revolver, or inherited one, and the action is stiff, the cylinder won't spin freely, it really needs to be detail-stripped, cleaned and lubed. There are books and web pages that cover this, or go to a friend who knows how(actually knows, not "Sure, I can figure that out") for help. There are a number of pieces inside, including springs
Note: if you have a Taurus revolver, you can remove the cylinder from the frame the same way, BUT in the end of the crane that fits into the frame, there is a plunger and spring. If you don't put them back in before you reassemble, the cylinder will not lock up. And they're easy to lose.