I understand your apprehension about owning and keeping a gun, however, there are no great alternatives for eliminating those kinds of predators without one. You would have to be at extremely close range to get a shot that would do any mortal damage with a pellet gun. Traps can work for some of the smaller pests but you then have to deal with relocating them.
If you do not have any friends that have experience with guns and how to safely store them and use them, you can contact the NRA and find a location near you that can offer a training course. Despite all the flack the NRA gets from part of the population, they are a good organization and want to make sure people are responsible gun owners.
When I was growing up learning about any piece of potentially dangerous equipment, my dad always said, "Don't fear it, respect it".
I wish you luck with your varment problem! You have received some great advice so far from folks on here.