first, make sure they are in season, and you have a hunting licence. if you are planning on trapping them, a trapping licence. follow all those rules and regs.
First thing, i dont think you are going to rid your yard of rabbits...they breed like rabbits and will be back. Instead of trying to kill all of them, perhaps just keep a handle on your dog so it dosent eat the droppings.
Next they are easiest to see in the snow, look for their eyes instead of the whole rabbit. and they tend to come out around dusk but will be active during the day. if its a real heavy snow you wont see them for a few days they tend to stay put. also even mid day they will be out sometimes sucking up some heat from the sun. Shotgun i suggest about 1oz or 1 1/8oz of #6's. Rifle a .22lr hollow point works great. As for trapping them i would suggest live traps so you arent killing an unindended animal. then just pop em in the head w/ the .22lr. They taste great so please use them not just kill to erradicate and disguard the bodies.