Better pull it completely apart and give it a thorough cleaning. Pull the buttstock by removing the recoil pad, then a 9/16 socket on a long extension or two together to remove the stock bolt in the bottom of the hole. Remove the foreend and the barrel. You should have 5 parts now; pad, buttstock, action, barrel, and foreend. Flush the action real good from every angle and in every opening you can find with brake cleaner, then quickly follow up with some good lubricant, I use Tri-Flow, but any good one that leaves a teflon coating will work. Let it get alll over the receiver and wipe it off good as this will restore the lustre that the brake cleaner will dull. Next clean and polish that barrel! Use Sweets 7:62 or another good amonia based solvent first, then make 50 passes with some Flitz or JB Bore Bright (not Bore Paste). Then flush it out with some Outers bore scrubber or similiar product, dry patch it really good and then Sweet's it again, and then dry patch really good again, and then apply a light coat of Rem oil or good gun oil. Something else I do, that I've been doing since I was a kid, is to apply a coat of Vaseline to all the metal surfaces, worked 30 years ago and still works now.
Sounds like a lot of work I know, but it will be worth it when you go to the range for the first time. Don't forget to bore-sight your scope beforehand too. You have purchased one the most infamously accurate shooters in a Handi Rifle so feel confident. I'd stay away from those 60 grainers to start with though, probably will shoot them after a 100 or so rounds tho. Good shootin' and keep us posted.