Not sure if this is the answer you're looking for, but Plutarch records Alexander the Great leaving his army to visit the site of the battle of Troy. Here he poured libations and inspected several weapons and relics of the Trojan war. Alexander apparently took a sword said to belong to Achilles and replaced it with his own.

We have records of individuals going all kinds of places pretty much as far back as we have writing, as MarshalOfNewZealand pointed out with Plutarch.

In terms of large scale tourism, that seems to have emerged with the Roman Empire. Apparently Sparta became quite a popular destination for well-to-do Romans who could make the trip. Once there, they could witness "authentic Spartan rituals," and eat in "authentic Spartan fashion." Apparently this even led to a revival of the culture of the Spartan golden age, the Agoge was re-instituted, although since Sparta was a part of Rome, it seems likely that it was only so wealthy foreigners could watch people beating each other up.