So I am going to assume you mean going on vacation for the general population, and not just the aristocracy (they've been taking it easy since forever!).

I only have information on the rise of vacations in France however. In 1936 France elected a left wing coalition (the "Front Populaire") which set the basis for social security in the nation. Among all the new 'benefits' awarded to workers were 2 weeks of paid vacation.

This marked the country's start of mass tourism, which coincided nicely with the expansion of France's railway and road system. Workers were now able to vacation to the beach, mountain or countryside. This also lead to a strong development of camping as a recreational activity - many of the workers could not afford hotels.

I would assume the advent of mass tourism follows the same trend in each country.