Welcome to the Forum Joe!
We do try & be polite here, as long as the folks who visit are polite as well. If you look around, you will see excellent posts by others about how this place is.
And in the spirit of that, I'll offer a bit of help.
First, the Ruger MKIII and even it's predecessors can be tricky to re-assemble for folks not familiar with them. Disassembly isn't as hard, it's getting things back in the proper place in the correct sequence that causes most headaches. So, if you take your gun apart, upon re-assembly, go slow, don't force the parts, (unless it's the receiver to frame fit that needs a bit of extra umph to seat it properly,) & stop & ask questions before messing something up. We will help when we can.
Next, since you didn't put a location in your info, I don't know where you are. BUT,,,,, if you do divulge your location, you may find we have a member nearby who can offer hands on help. I'm in Western NC, and my door & range is open to you.
Now, what to clean.
Some here never disassemble their MK series autos. They will remove the grip panels, (if they have them,) spray a brake cleaner all around the insides letting it flush out the gunk inside things. Wipe it off, apply a LIGHT lube & be fine. Others do the field strip as mentioned in the manual, and clean things from there.
If you do the field strip, I'd use the brake cleaner spray for the lower grip frame assembly insides. It's easy & very effective. Add a light oil to the mating & moving surfaces & the lower is good to go. An old toothbrush, (or similar brush,) can also allow you to brush out any crud you may need to remove. Next the bolt. A good cleaner will remove the surface gunk, and again, a toothbrush can & will clean the boltface. Spray the extractor area & move it a lot to remove the gunk in that channel.
The receiver, clean it inside. Especially the face area where the bolt closes to it. I use a toothpick or similar object to "dig" out the gunk in the corner all around the circumference. One area that needs close attention, is the extractor slot in the receiver. Again, a toothpick is an excellent tool to dig out that space, as well as a good spraying.
Lastly, I use a boresnake down the bore, or a wet patch on a rod jag, followed by a proper sized brush, followed by a few dry patches.
That's all I do to clean mine.
I hope this helps!