**revision** Looks like Ben beat me to it, but here is the long winded version****
Before you do anything to the rifle, SHOOT IT! The only way to see what it needs and how it fits your needs and shooting style is - SHOOT IT! I see no need in tinkering with something that doesn't need it. I got my first handi and it was a process for me. I did not have the great support of GREYBEARDS back then to help me through it, but I found there wasn't much I needed to do to the rifle for it to shoot. I shot about 100 rounds through it before I felt like I was competent with it, no fault of the rifle.
So my advice is - SHOOT IT! CLEAN IT! ENJOY IT! If it still needs something to make it your own or to fit what you want from it, somebody here can help you, but right now you don't know what you need or want!
When I got through those 100 rounds I found that whenever I cleaned the barrel with copper solvent, (something that I read in a magazine at the time, I already cleaned it after every range session just added the copper solvent) it no longer came out green, but mostly black with a little green.
I don't think it had as much to do with the accuracy as me learning the rifle and practicing shooting with it.