Here's a link to the official AZ Bike Map
MAG|Bikeways

And the Strava Heatmap is a great way to find routes the locals use.
http://labs.strava.com/heatmap/#12/-...5300/gray/bike

Depending on where you're staying in Phoenix and how long your bike rides are, you might have the same success with the GRID bike share system as you would with renting a commuter.
gridbikes.com

You'll be fine riding in the evening or the morning. The times you usually want to skip are from 2pm-5pm which is the hottest part of our day. It's still going to be hot when you're riding, but it's not that bad - there's always a breeze on the bike and as long as you're drinking water and staying hydrated, you shouldn't have any issue.

Just east of Phoenix is the Coon's Bluff campground on the Salt River, known for it's wild horses. It's a free campground with just bathroom facilities and water and a lot of people use it to push off from while tubing down the river, so it can be a bit trashy, or it can be super nice. We camped there in March and the place was empty. We were right on the water. There are tons of hiking trails in the area and mountain biking nearby as well.
Tonto National Forest - Coon Bluff
Coon Bluff Campground on the Salt River

Unfortunately, I haven't done much camping near Phoenix, so I'm not much help. Sedona is only a couple hours away from Phoenix though, and I'd say that would be worth the drive if you haven't been.