I am not the best source of information because I only rarely drive all the way across Tucson (I live on the far west).
There is a veritable maze of trails in the desert around the visitor center. There are trailheads near the east ends of both Speedway and Broadway that immediately enter the park. I am told that the letter of the law says you should be paying for an entry permit if you use these, but this is never enforced (at least not as of early 2010). You will want to refer to this map of the low trails in Saguaro National Park (from the park website), stashed here for your convenience.
The Douglas Spring Trailhead (east Speedway) is at approximately 2700 feet elevation. Douglas Spring is at 4800 feet, 6 miles and 2100 feet of climbing. This makes a vigorous day hike for any time other than summer. The park service brochures call this hike "strenuous", yet indicate that you can do the 12 mile loop in 5 hours, which is probably true for someone in very good shape.
Tom's hiking pages / tom@mmto.org