Monsoons begin in July through September - usually.
Snowbirds head for home the end of March to do their taxes. :)
Best months in spring are February through April. May and June are very hot.
Can you fit in Tucson and the greater Phoenix areas:
For the greater Phoenix area we'd highly recommend Lost Dutchman State Park in Apache Junction. You have a big choice of sites for dry camping to electric to full hookups. It's a awesome park for spring flowers beginning in February for the hillsides of poppies. It has very nice hiking and ranger programs and there's lots to do in the area. We've volunteered there. :) You could bike down the Apache Trail nearby for a beautiful experience with lakes and terrific scenery.
Other very nice campgrounds are through the Maricopa County Parks. They are scattered around the greater Phoenix area. They have lush, big sites and great hiking. They will have electric. Favorites of ours are Usery, McDowell Mountain and Cave Creek.
For the Tucson area Catalina State Park is terrific and has electric. Another nice county campground is in Tucson Mountain Park - Gilbert Ray campground.
If you want a lot of activities the Voyageur RV Park in Tucson is the best.
For mountain biking any of these parks would be good. McDowell Mountain - a Maricopa County park campground would probably be the best for the many trails within the park. It's in Fountain Hills. For Tucson, Gilbert Ray has good trails but Tucson as a whole offers many biking routes if you pick up a map.
As you can tell, these are all favorite areas of ours. Enjoy!