Stack luck (Ginger Ale and Lucky Lunch/Magic Rock Candy) then go panning on Ginger Island to get your first lucky ring. If you don't succeed take your stacked luck to Skull Cavern and just do a normal run and you might get one as a drop. With high luck your chances of getting an auto-petter and auto-getter from the loot floors. I have two lucky rings and always do ginger ale and rock candy and my luck stacks so high that I get pretty much a 100% drop rate of the things to the point it's kind of a nuisance. Once find a lucky ring put it on. Later on you can take it to the forge to fuse with a burglars for double drops and again Iridium for the magnet. Now every day is a good luck day.

EDIT: You can also combine lucky rings with your favorite set of rings. For example luck/coffee with luck/burglars is a very funny combo.