Backpacking Arizona

My girlfriend and I are planning a backpacking trip to Arizona from April 17-24. Got a last minute deal on cheap airfare from Missoula, Mt, so we're kind of winging it. Researched online... tons of info but you can't beat firsthand knowledge. If anyone has any suggestions I would greatly appreciate it. If you could spend one week backpacking in Arizona where would you go? We would like to dodge the snow for the most part..... it's been a long winter here in MT! We would also like to dodge the crowds, if possible. From our years of experience we have had the most enjoyable times in wilderness areas, so that would be ideal. We've never been to the southwest so I want to play it safe the first time around. We don't want to put ourselves in any sketchy situations since we're unfamiliar with the terrain; maps and photos can only provide so much insight. Any suggestions or tips would be appreciated.