Ruger mark III or 22/45 can be found pretty cheap used. Great pistols! Ruger 10/22 Carbine is on sale for 179 fairly often.
think with the 10/22 you can't go wrong. there is a ton of aftermarket parts to build it anyway you can imagine. I have found that th stock barrel can very very accurate if you give it the chance. I have heard that accuracy out of the stock barrels can vary. I think most people don't give them the time. i think it takes about 500 rounds to really break them in before you can judge their accuracy's. That being said for plinking time they do what they are made for. Now if you want a one hole scoped tack driver they can be built to do that. I for one stick with iron sights. i think learning to shoot with irons at distance 50 to 100 yards is critical to good marksmanship.
As for a handgun, again the ruger markII, their 22/45 is another great choice. I went with a 22 lr conversion for my go to weapon to get more trigger time. Now thats if their is a good conversion kit for you weapon. Again, i am not looking for a competition weapon just something i can have fun with, practice my marksmanship and get great practice with sight picture and trigger control.
Get a better scope and some decent rings and it might surprise you. My Marlin (Model 60, but I think the barrels and much of the action is the same) is capable of hitting golf balls at 100 yards more often than not.