I expect the lines will be blurry at best, but if we think of "tourism" in the sense of travel undertaken mainly for the purposes of entertainment/relaxation/etc. and not motivated by religious, educational or professional interests.
The tourism era began shortly after the advent of money by the Sumerians around 4000 BCE. This was the dawn of modern "travel" as we know it, however the industry wouldn't become anything we'd recognise today until thousands of years later. "The first journey ever made for purposes of peace and tourism was made by Queen Hatshepsut to the land of Punt ... in 1480 BCE" (Goeldner and Ritchie 2012).
Tourism became more familiar during the era between the dates listed above due to the construction of large burial tombs by the pharos beginning in 2700 BCE which drew many visitors. The Egyptians also met several times a year to participate in large festivals which also gleaned visitors from within Egypt and the surrounding areas. The largest festival at Bubastis drew (literal) boatloads of people each year.
Personally, I would credit the Egyptians as the fathers of modern tourism. The relics and monuments we visit today were just as desirable a place to visit thousands of years ago.
Tourism expanded as nations built roads and navies. The Greeks were building paved roads during the same time as the Egyptian empire's expansion, and the romans finally brought it all together with their widespread road construction around 150 BCE.
Hope this answers some of your questions, if you're curious about any details, I can dig my nose in this textbook some more and hopefully answer then more completely.
What, in view of this question, counts as tourism? Does the Islamic Hajj count? Tradesmen on the Silk Road? Or is tourism solely for pleasure or to expand ones own horizons?