If you're going to go exploring in the Tortolitas, do it right and ride responsibility.
Buy a Recreational Permit to ride on Arizona State Trust Land. It's easy. Just call 628-5480 in Tucson and ask. The individual permits are $15/year, or were in mid-2001. The folks there a very helpful, very knowledgeable, and usually get your application out the same day. Permit fees help keep trust land open to public use. Do it to support continued mountain biking access to these lands. If that doesn't work for you, do it because it's the law.
Follow the rules. The permit allows you to use trust land within defined limits. Check out the booklet when you get it. It does not allow you to cut trails, cut fences, harass livestock, 4-wheel or mountain bike off-road, or have a party, among other things.
No Trespassing signs mean just that. Respect private property. You don't want strangers in your backyard cutting down fences and chopping down trees, so stay out of their's. It's not only respectful, it's the law.