Wealthy Romans travelled to certain places very often, towns like Pompeii and Herculaneum in Campania were greatly supported by what we might call a tourist industry. The region is fertile, so there was a lot of agricultural produce, but because of the amount of holiday villas in the region, this produce aimed at luxury consumption -- wine and fresh fruit.
There was definitely an industry that thrived on the tourism of the wealthy, but the tourists themselves were not travelling for pleasure in the same way that we might now. In Cicero's letters, for example, he discusses with his friends where he will spend winter, but most of his thoughts seem to be political rather than about places he and his family want to go.
On another level, there are also examples of travellers doing so just for pleasure and to see their world, Pausanias and Strabo for example give hugely useful accounts of their travels. The traveller is also a stock character in many tragedies, comedies and later on in latin novels, so it seems that travel was (obviously) relatively widespread even if it's not strictly tourism.
This last idea is only vaguely related to your question, but you should look at this search engine staff at Stanford have been working on. It gives you an estimate of how expensive and how long simple journeys were in the ancient world. I was hugely surprised by how quickly trade ships etc could move around, and this comes of the back of a wave of scholarship that stresses the interconnection of the Mediterranean world across what we might expect to be rigid national communities. So, certain travels were widespread but the extent to which people travelled for pleasure is difficult to ascertain.