Remove the screw from in front of the trigger, GENTLY swing out the cylinder and remove the cylinder assembly. That should be all the stripping you need to do.
Get a Lewis Lead Remover for the bore.
There's no good reason to remove the cylinder assembly for routine cleaning - leave
it and the sideplate alone.
As to the Lewis Lead Remover - if you're not shooting lead bullets, there's no need for it. Even if you are shooting lead, unless you're having a problem with leading, you ought not need more than occasional use.
IMO, people spend way too much time cleaning their wheelgun, trying to get it looking new. At best, it's unnecessary. At worst, you can end up damaging or adding wear to the gun. The front face of the cylinder, in particular, seems to bring out peoples' obsessive side. Just wipe the heavy stuff off, then let the rest be. It'll just come back, and getting it like "factory new" is a losing battle.
After shooting, wipe the gun down with solvent. Get the stuff between the forcing cone and the top strap. Then run a solvent soaked patch through the bore (preferable from the breech end) and chambers, followed by a dry one. If you shoot .38spls through a .357mag, occasionally run a brush through the chambers. Even here, it doesn't need to be done every time you shoot .38spls.
Every so often, I'll use a brass brush & solvent on the front of the forcing cone (and in the space under the top strap) as well as on the front of the cylinder, the cylinder flutes, the cylinder stop slots, and on the ejector star on the back of the cylinder.
Anything else to be aware of?
I'm surprised no one mentioned this: When shooting .357mags through any k-frame .357, stick with 158gr, and avoid 125gr ammo.