The GI multi-piece rod and kit is handy if out to the range, or in the boonies.
Coming home from the range, I like having a one-piece coated (or synthetic) rod. I keep a jag tip; and both bristle and loop brushes; and I have a dedicated mop tip, too. I have a dedicated chamber brush on a flexible rod with a T-handle.
As to method, that's simple enough.
Clear weapon.
Pop front & back pins, and separate upper and lower. Remove BCG & charging handle.
Take down BCG and apply cleaner & set aside.
Flush gas tube from breech with gun scrub.
Grub out chamber with wet brush.
Wet patch down bore.
Bore brush bore.
Wet patch again, then dry patches until clean.
If it's been a good stretch, I'll wet the mop tip in tetra and run that down 5-6 times.
I'll then bore-snake the tetra in.
After that, it's just cleaning & lubing the remaining parts and reassembling.
Ok, I just looked, I have another bore-snake in my Waller AR bag. That one is "skinnier" than the one in my multi-drawer cleaning box.