A couple of points to keep in mind. Be wary and mindfull of your neighbors, last year one called the cops on me when I was in the back killing pigeons who were crapping all over my tomatoes. The boys in blue showed up with their pistols drawn because they were told that some nut was outside shooting a gun randomly. I almost spent a few nights in jail but I convinced them it was only a pellet gun and I was only doing pest control.
Some birds are easier to kill than others. Robins are pathetic, they can be killed flat by spitting on them. Pigeons are tough little birds, you need a head or spine shot to drop one. A pellet in the belly will eventually kill them but it takes a while. Squirrels are suprisingly tough. I find them harder to kill than possums. Skin one and you'll see why, nothing but bone and mussle. I prefer to use .22LR's or #4's in a .410 shotgun.
If you do use a .22LR and worry about your neighbors getting upset about the noise then go with the subsonic rounds. They're dirty little rounds but in a rifle length barrel make very litlte noise.