Are you there for a specific team? Will you be mostly in the East Valley (e.g. Scottsdale)? The only trail riding I've done was at McDowell Mtn Park and Tonto National Forest (both north Scottsdale). Papago Park I believe has trails and is fairly close to lots of the baseball parks. Usery Park is in Mesa and has some good dirt. I hiked the trail around the mountain (7 or 8 miles)...parts of it would be a bit scary to me on a bike, plus it's pretty popular so you'd have to yield to hikers quite often. If you take a road bike, riding out to Saguaro Lake is a nice trip. It's a great time of year to be there...enjoy!
