Give me an idea for a country to travel to that is safe, inexpensive and does not fall into the well trodden tourist path?

One such place is Mongolia. Situated between China and Russia, it is an ideal destination to explore national parks, deserts, and history. Mongolia is the historic home to one of the largest empires to ever conquered the earth. The Mongol empire covered most of Eurasia and had a fierce nomadic culture and modern weaponry. 
As a traveler to Mongolia, you can spend time with today's nomadic tribes, hike through the Khstain Nuruu National Park park while sleeping in luxury tents, marvel at the beauty of Ugil Lake, or Camel ride through the dunes of the Gobi desert. 
You will enjoy Mongolia while feeling safe and awed without breaking the bank.