The Indian Springs Trail leaves the Rainbow Campground (by site #108) on the south side of Big Lake Arizona. It's west of Alpine by about 15 miles, about the same distance South of Eager. It's not a trail you'd travel 100 miles to ride, but well worth hauling the mountain bike along with you on that family fishing trip to Big Lake.
There are a couple of additional miles you can do by a) climbing up to the Big Lake look-out tower, which is very steep towards the end but rewards with great views, and b) doing the West Fork trail which spurs off from the Indian Springs trail. This one will add 7 miles total and is an out-n-back ride to the black river. It's well worth the time. (Or just do multiple loops of Indian Springs).
Overall, the trails are rolling, non-technical affairs with mixed woods and 'alpine meadow' riding. If you're from the desert (like Tucson) you're going to love the change of scenery and fast rolling single track through............no, not cactus..........but real trees!!
So haul the bike and enjoy some great riding. It will certainly help you forget how much you stink at fishing.