I've been lurking on these forums for a while but saw this question and decided it was time to weigh in. So hello, everyone.
Mark,
Hunting rabbits is a lot of fun and pretty simple. In southeast Arizona, depending on elevation, you'll be chasing either eastern cottontails or desert cottontails. I currently live in Oklahoma where we hunt eastern cottontails, but I grew up in Southern California chasing desert bunnies. The main difference that I've seen is that desert rabbits tend to use burrows more. Once they go down the hole you're not likely to see them again, and they spend most of the day down there out of the heat.
So step one is to grab your 12-gauge and some #6 or #7.5 shotshells. Use an improved cylinder or modified choke. Your rifle is way too much firepower for bunnies; look at them hard and they fall over dead (unlike squirrels, for instance). Step two is to find some land with plenty of brushy cover. Out here a sure bet is an old overgrown pasture choked with blackberry brambles - the more miserable it is to walk through, the better it is for hunting bunnies. I'm guessing that you'd do well to look for sagebrush and cactus. Desert rabbits don't need water but they do need fresh green vegetation to eat so keep that in mind. If you can get permission to hunt a farm, look for weedy edges near crops. Rabbits love that stuff.
Rabbits use the same paths repeatedly and beat down paths into the thick cover, so look for these trails and rabbit pellets along the edges of brush. Also look for large rocks with gaps underneath that they can squeeze into. South-facing hillsides seem to produce more bunnies than north-facing.
After you've identified some likely cover or have actually seen some rabbits while scouting, it's time to go hunting. All rabbits are most active at dawn and dusk and you're best off hunting the first and last couple of hours of light. Walk along brushy edges and stop to kick likely cover - often rabbits will freeze in place as you're approaching and will seemingly materialize between your feet, tearing off for the nearest patch of brush. You have to be quick on the draw - if there's a hole nearby they'll head right for it and you have to hit them before they disappear. On the other hand, if a rabbit doesn't have a hole to duck into it will often run about 10-15 yards away and then freeze, hoping you can't see it against the brush. So sometimes you get two chances.
Good luck and have fun. You'll figure out what works best for you - I have a buddy who moves quickly in an effort to cover as much ground as possible, while I prefer to stop frequently to make any nearby rabbits nervous that I'm onto them. Both methods work well. I enjoy hunting rabbits because the season is long (year-round in AZ, I think), they're very tasty, and you don't need a bunch of expensive gear to be successful. Just a shotgun and a good pair of boots.
