The Dragoon Mountains in southeast Arizona are a great mid-altitude range, and are pretty easy to get to. The Cochise Stronghold Trail is a nice combo of distance, difficulty (the elevation gain is only about 1100 feet - but you climb up the divide each way so double it up), and uniquely awesome scenery.
Trailhead: Take off from the campground on the east side of the divide - there is also a western approach but it's 4-WD. There is a signed takeoff for the Cochise Trail. It actually begins from a short nature trail.
Route: Start by crossing a bridge and immediately coming to a fork (the nature trail is a loop). Take quick left and follow for a couple minutes to a sign pointing you off to the left again. Again follow for just a couple minutes and a horse trail joins from your left and together (as the sign says) you take a right. Now you are off...
Immediately the scenery is awesome and all around, huge boulder-laced mountains. You start to slowly climb along a wash for about a half-mile before coming to a hiker's gate. Pass through and continue your ascent, which is very pleasant. Just short of a mile you come to a kind of clearing, with some the floor of some sort of really old structure still left in the ground. Hang a right here (there appears to be a more faint trail to your left) and follow up the wash for a while longer. The flora here is cool, with manzanita, juniper, and small pines growing alongside yucca/agave and prickly pear. Also there is a huge amount of beargrass lining the way. Eventually the trail swings right and heads out of the wash (which is never that far away as you'll see). As you climb up more the trail takes on more of a cross country feel and becomes more rocky. The trail is pretty easy to follow, anytime you lose it just look for well placed cairns and remember that you are always heading west.
The trail climbs up a bit more then opens up and becomes absolutely awesome. Boulder-laced mountains fall away into the distance, many with massive bald domes of rock. This part of the walk is incredible. At about 3 miles you come to the Stronghold Divide. You can turn around or head down to the other access point on the west side.
The hike down is straightforward as it switchbacks down the canyon. The views to your right are always tremendous. After about 1.75 miles you ultimately end back at ground level at the western trailhead - which looks to be pretty remote.
Turn around and start your climb all over. The way back is tougher as you regain all the elevation in a little more than half the distance, but it's not that bad. The views are straight ahead the whole way and that's pretty good incentive.