First, get with a friend or relative to get some hands on training

they are quite an experience, especially if you've never had a gun buck in your hands. Once you do get one, go back to the same person and get them to walk through with you loading, siting, firing, etc until you are completely comfortable.
A 410 shotgun will shoot shells or slugs (BB type shot & regular bullets) and should be sufficient for most yard pests. I'm sorry, but I can't recommend a brand. The one I have is an antique Winchester.
I am partial to my handgun, a Springfield 1911 .45 ACP. It will take care of most anything, but it would prob be overkill for minor pests. You can get "training" barrel kits that shoot a .22 instead of the massive .45, but they will run a pretty penny.