Tecolote,
Most recommend that you remove the grips (one screw in the back strap), soak the gun in Gun Scrubber, wipe off, coat with CLP, wipe off, and then lube and assemble. Personally, I just removed the grips, coated the outside of the gun with CLP, let soak a little, than wiped off the grease. After the outside was clean, I then disassembled the gun, and cleaned the inside the same way, but I did go through a lot of paper towels. The serrations on the safety and hammer have some sharp edges, and it's easy to see how you cut your thumb. I'll be touching up mine with a file soon.
Disassembly of the gun is easy. This pistol has a fixed barrel that is part of the frame. The trigger guard is hinged. All you do is remove the magazine, take the gun off of safe, pull down the front of the trigger guard (it should lock in place), pull the slide toward the rear and upward until the slide clears, than push the slide forward off the barrel. The recoil spring loops around the outside of the barrel, and can be removed by turning it (kind of like unscrewing it off the barrel).
One of the first things you should do, is replace the stock plastic grips.
www.markarov.com has Pierce, Makawrap, or wood grips at reasonable prices. I just ordered the Pierce. You should really check out that site, as it's loaded with all kinds of information on these pistols.
Good Luck,
David