There certainly were. Although tourism as we would define it was a less common reason to travel long distances than commerce or pilgrimage, a substantial number of wealthy Romans spent months or even years exploring far-flung corners of the Empire.
The most famous tourist was of course the emperor Hadrian, who spent nearly half his 21-year reign wandering the provinces. There were good political reasons to do this; like so many premodern rulers, Hadrian knew that there was no better way to win local loyalties than to tour his domains. But he seems to have always been motivated largely by pure curiosity. Among many other exploits, Hadrian exhausted the scholars at the Library of Alexandria with questions, hunted bears in northern Asia Minor (Turkey) and lions in Libya, sailed up the Nile, and climbed Mount Etna to watch dawn filter through falling ash.
Although Hadrian was exceptional, there were many other Roman travelers. The default summer resort of the Roman elite was the Bay of Naples, with its exotic volcanic phenomena, Greek-inflected culture, and easygoing morality. Many Roman notables, however, at least occasionally visited more exotic locales. The most prominent "cultural" destination, as you might expect, was Greece. When they were young, many elite Romans spent the equivalent of their "semester abroad" in Athens, perfecting their Greek and pretending to listen to the lectures of famous philosophers. Some returned when they were older, occasionally (if they had no political interests to bring them back to Italy) for years at a time. From Athens, Roman visitors made forays to many other parts of Greece. Sparta - where a grimly glamorized version of the traditional military training was performed for tourists - was a favorite destination, as was the great shrine of Asclepius at Epidaurus (the Empire's most fashionable healing sanctuary).
Some Romans continued east to the great cities of western Asia Minor (now the Aegean coast of Turkey). The site of Troy, which the Romans called Ilium, was a favorite destination, since the Romans liked to imagine that they were descended from the Trojans via the hero Aeneas and his followers. Other visitors made their way to the famously beautiful city of Smyrna, to the immense Temple of Artemis of Ephesus, and to the gorgeous Mausoleum at Halicarnassus.
Probably the favorite tourist destination after mainland Greece, however, was Egypt. In Alexandria, where ships from Roman and Corinth docked, visitors could marvel at the Tomb of Alexander the Great, at the towering lighthouse, and at the mile-long colonnades along the great boulevards. Then, like modern passengers on a Nile Cruise, they headed upriver. The Pyramids were a perennial source of amazement (before medieval rulers tore off the smooth limestone casing, the blocks were covered with Greek and Latin graffiti), as were the immense temples and tombs around Thebes. The so-called Colossus of Memnon, an immense statue of Amenhotep III that emitted a high-pitched sound when struck by the first rays of dawn (probably water escaping from cracks in the stone), was Egypt's biggest tourist trap. For a fee, Romans were even allowed to have their names carved into the legs and feet of the statue. Many did - including Hadrian and his entourage.