Basically, you clean a) moving parts, and b) parts that are in contact with the exploding shell. Don't over-oil things like springs, because that will attract and catch dust and dirt, and muck them up. Put a fine sheen of oil on everything to help fight corrosion, but don't leave it dripping. But yes, the cleaning processes are the same regardless of brand.
As for cleaning procedures and materials/tools, go ask your local gun shop for help. I have yet to find a store that isn't happy to answer questions like this, and give pointers to help keep shooters safe, and keep the hobby fun.
My cleaning kit is a bore brush; cotton cloth pieces to run up and down the barrel; a bore "sponge" (I'm sure that's not the right word for it, but it's an absorbent piece to get cleaning fluid into the bore to loosen firing residue); an old toothbrush; a larger cotton cloth for wiping stuff down; a bottle of Hoppe's cleaning fluid; and a bottle of gun lube for final assembly.
After you fire your gun, you'll see firing residue on certain parts. Clean it off, use the toothbrush if you can't reach it by hand, wipe it down, give it a light coat of oil, and reassemble. You'll learn your gun's personal geography after a couple times through the process.
Welcome aboard!