You are visiting t  tricky time of year, even at low elevations, since storms are likely.  Sometimes these storms leave snow on the ground in Tucson.  But there are lots of possibilities - Saguaro National Park, both Est and West units, is a nice visit, but I am not sure about dogs - they probably must be on leash. you also have the Tucson Mountains and the Arizona-Sonora Desert Museum.  Developed caves tours are available on stormy days.

Picacho  Peak has a developed trail to the summit which requires only care, not any technical climbing gear or skills.  Your pooch can probably handle the trip.

You can go higher and deal with as much cold and snow as you wish, but conditions can vary from year to year.  Pay attention to the weather forecast.

It has been a while since I have lived in Tucson, so I am sure people will respond with more current info....