To confirm comments upthread, during my nine years of military service ending in the mid 1980s, a military ID and leave orders were all one needed to enter NATO countries, Australia, New Zealand, and other allied nations.

The U.S. still has some good recreation bases in Germany (Interlaken ?), not to mention the Hale Koa on Waikiki, etc. The active duty folks should check out their Welfare and Recreation office. There are travel books sold at Army and Air Force Exchange Service stores and Navy Exchange stores geared toward military folks.

In addition to commercial flights, they can catch space available flights (hops) with the Military Airlift Command, though it is tough during summer with all the PCS (Permanent Change of Station) family moves, who have priority over personnel on leave. Military bases throughout the world also serve as de facto hotels, with the BEQs (Bachelor Enlisted Quarters) and BOQs (Bachelor Officers Quarters) -- again, leave status having the lowest priority. Some are in quite nice locations. I made good use of this fringe benefit, also flying on reserve P-3s going transcon.

For many lower and middle class folks, the military is indeed their ticket to see the world.