There are some problems with terms such as "tourism" and "vacation." Since, after all, what about a relaxing stroll down the street or a visit to a relative in another town?
Nevertheless, modern mass tourism in the West started in the early 19th century. For the UK, some say it started after the Napoleonic Wars ended, when middle class British were free to travel in Europe. This is, after all, the time when tourist agencies arose and travel guides began being published. Karl Baedecker's guides began being published in 1828; John Murray's travel guides began being published in 1836.
This type of mass tourism was still based on an earlier form of tourism that gentlemen of the wealthy elite in Europe practiced from the 16th to 18th centuries called the Grand Tour. The purpose was to educated them, gain knowledge about other cultures, and also to prepare them for diplomatic careers. They often wrote books about their travels upon return home, writing such books as Richard Lassels' Voyage of Italy, published in 1670.
So, early mass tourism in the 19th century largely followed this model, and as such, can be called "cultural tourism," which of course still exists today. As for sun and beach tourism, which has little interest in culture or edification, that started later. However, one would also need to keep in mind that visiting spas and other such resort spots in Europe is not new. Also, one needs to consider that people have been traveling since the beginning of time and they have written accounts of those travels. To what extent were Marco Polo or Christopher Columbus being tourists?
EDIT: typos
The "Grand Tour" was an aristocratic young man's "right of passage" into adulthood. It involved him traveling to Italy and France to learn about art and antiquity. It started around the 1600s.
Another question: what about pilgrimage in the middle ages? Was that an equivalent to tourism?
You may be interested in some of these previous threads about tourism and vacations:
When did traveling for pleasure become popular?
At what point in time did traveling for leisure become normal?
When were vacations (as we know them) "invented"?
When and where did tourism start?
When and why did people start leaving home for short vacations?
When did tourism become a large part of countries incomes?
Did people use boats and ships as pleasure craft before modern times?
(Looks like it's time for a new section in the Popular Questions pages!)
EDIT: Here we go! Vacations, holidays, and tourism
In Germany there had the Wandervögel (migratory birds) - a movement of young people that was all about romanticism / tourism (since 1896)
The English wikipedia entry says that some authors found it similar to the hippies, i would say that they would have been more similar to scouts.
I just wanted to chime in on this real quick and maybe point out that some of the early forms of "tourism" may have been reverse tourism. Sara Baartman is one such example moving from South Africa to be exploited as an untammable savage choi choi in London and later Paris. As such Savage Africa (at least the european understanding of it) became accessible to usually nobility. This gets more directly linked to your question in the USA where aboriginal Australians (and other peoples) would become sideshows to the Barnus and Bailey circus company and would travel to middle class america c.1900. If you care to read more I will find you a link later but currently my internet sucks and the book I read it in is at home.