If so, what sort of places would they visit? How long could they afford to stay? Would they take food with them to save money? Would single women take vacations and travel alone?
Edit: One more question--what kind of accommodations could a lower or middle class person afford?
No, they went on 'holidays.' Joking. Yes, they did. During the 19th century, the rising middle classes and working people alike found themselves with increasing amounts of leisure time, not least as labour unions and increased political pressure brought working hours down. Add to this the advent of regularized and affordable rail travel, and the modern 'vacation' was born.
Perhaps the most popular destination were seaside resort towns. Rail companies were always looking to increase ticket sails, especially on weekends and to the ends of long rail lines. Why not build a resort town or pleasure park at the end of that rail line to boost ticket sails? Think of places like Brighton or Blackpool, where the Victorian masses could promenade along beautifully decorated piers, purchase cheap confections or food, and ride a new roller coaster or carousel. They might even stay overnight in a relatively affordable hotel or guesthouse, many of which still survive in Britain's seaside destinations today.
More distant travel, however, remained an expensive luxury. The upper classes and the wealthy, aspiring middle classes had long traveled through Europe and to the holy land -- the 'Grand Tour' was a staple of 18th century gentlemanly education. However, as the 19th century gave way to the 20th, regularized steamers looked to boost their tourist trade, and lines like P&O brought ever larger numbers of British passengers to destinations in Europe and further afield.
Much of that nascent travel industry targeted not single women or men, but families. But one can see anxiety about sex and gender in much of the period's travel and holiday activity. Bathing at mixed-gender beaches was a sanitized affair; during the middle decades of the 19th century, it was understood as a salubrious health-cure as much as a recreation. The Victorian bathing suit is a stereotype, but it wasn't until the twentieth century that fashions generally became more revealing. An unaccompanied single woman would have faced some stigma, though that did not stop some of those young, single, working women with disposable incomes, especially in urban centers, from enjoying leisurely pursuits.
A few books and articles from a growing discourse on the history of leisure and travel:
Peter Bailey, Leisure and Class in Victorian England: Rational Recreation and the Contest for Control, 1830-1885 (Routledge, 2014).
Judith Flanders, Consuming Passions: Leisure and Pleasure in Victorian Britain (HarperPress, 2006).
Pamela Horn, Pleasures and Pastimes in Victorian Britain (Amberley Publishing Limited, 2013).
John K. Walton, The British Seaside: Holidays and Resorts in the Twentieth Century (Manchester University Press, 2000).
Parratt Cm, “Making Leisure Work: Women’s Rational Recreation in Late Victorian and Edwardian England.,” Journal of Sport History 26, no. 3 (December 1998): 471–87.