Travel was dangerous in the ancient world, so casual tourism wasn't as much of a thing. You might have another estate in the countryside (because cities were hell) if you were rich enough or you might travel for work if you were a merchant or sailor, but recreational tourism was too much of a risk to be very appealing. However, there is a strong tradition of religious tourism in the ancient world. A Greek might go to Delphi to consult the oracle or to Epidaurus to be healed at the Asklepeion or go to the games at Olympia. In these cases, you don't take a souvenir home with you, you leave something behind. So around a lot of sanctuaries were shops where a visitor could buy a votive offering - a small terracotta, a nice marble plaque, a little bronze figurine - whatever you could afford which you would then donate to the sanctuary. These votives are sometimes inscribed with information about the dedicator like their name, place of origin, something about their visit. While we have a lot of evidence for this kind of practice, there is little evidence to suggest that visitors brought back anything other than some new stories.

However, this does start to change in the Roman imperial period (which probably has a lot to do with the improved road networks and the centralized government). A significant change shows up in the religious landscape, people start bringing back souvenirs, particularly relics or water / dirt from places of healing.

Of course, these are more than tourist trash, they have real power in the minds of the pilgrims. If you're looking for the kind of gimmicky stuff that you see nowadays though, this also starts showing up in the Roman period. There are glass vials from around Naples that show the local sights. Travel in the Ancient World by Lionel Casson kind of oversells the idea of souvenirs (he doesn't really distinguish between locally produced goods that are traded and "souvenirs"), but fig. 19 and 20 give some good examples of these flasks and pilgrimage souvenirs respectively.