Cheyenne Mountain does offer tours, but not to members of the public. That program was scaled down in 1999 and discontinued after September 11.
TripAdvisor reviews from a few years ago suggest that "Friends and Family" tours can be scheduled on Fridays, but you must be invited and escorted by someone who works at the facility, and must be a member of the U.S. military or be pre-screened by the Department of Defense. I could not find screening information online, but for comparison, it takes 4-6 weeks to get approval to visit the U.S. Naval Observatory in Washington, D.C.— and unlike Cheyenne Mountain, that is a facility that does advertise public access. You may not be missing much; most of NORAD's operations have moved elsewhere.
The easiest military-related attraction to visit in Colorado Springs is the National Museum of World War II Aviation, located on the grounds of the cos airport and open several days a week. The Peterson Air and Space Museum is also open to the public, but Peterson Air Force Base where it is located is not, so you will need prior authorization.
Fortunately, there are many more easily-accessed places to sate your appetite for Cold War-era military facilities: the National Atomic Testing Museum in Las Vegas, Greenbrier Bunker in West Virginia, the Titan Missile Museum in Arizona, the Submarine Force Museum and USS Nautilus in Connecticut (adjacent to the Naval Submarine Base New London, which allows public access on certain days but doesn't offer much to see), and the Minuteman Missile National Historic Site in South Dakota.