It would depend in part on the kind of books you wished to access, but your main options would be:
to purchase inexpensive 'yellowback' novels from your local bookseller (assuming you lived in a city or sizeable town). Yellowbacks, so-called from their yellow cardboard binding, were cheap reprints of popular novels like Robert Louis Stevenson's 'Kidnapped'; somewhat more respectable than the 'penny dreadfuls', or sensational horror/thriller stores which retailed for a few pence. They were produced with ultra-thin, machine-made wood pulp paper, which was cheaper than the hand-made cotton rag paper used for fine volumes and expensive engravings, and its thinness meant that a long work could be incorporated into one volume, rather than a reader having to buy multiple volumes. They retailed around the price of one shilling - still a substantial investment for a working-class labourer on a weekly wage of 20 to 40 shillings, but just about affordable as an occasional purchase. The British Library has more information on yellowbacks here: http://www.bl.uk/collections/early/victorian/pu_yello.html
to visit your local free public library, often (at the very end of the century) a Carnegie Library, funded by the steel magnate and philanthropist Andrew Carnegie. The Public Libraries Act of 1850 provided the impetus for municipal library-building, especially in northern industrial towns such as Salford. There's a comprehensive guide to the history of 19th century British public libraries here: https://content.historicengland.org.uk/images-books/publications/iha-english-public-library-1850-1939/heag135-the-english-public-library-1850-1939-iha.pdf/
to join a 'Mechanics' Institute'; a kind of combined library and social club for working class people who wished to pursue education and self-improvement. Mechanics' Institutes existed in most major English (and Scottish) cities, proliferating in the 1820s and 1830s. In addition to the provision of a library (which, importantly, would often by lit by gas-lamps, allowing labourers to read after working hours in the evenings), they often provided public lectures and technical demonstrations. Many Mechanics' Institutes would subsequently morph into full universities by the late 19th or early 20th century - Birkbeck College being a good example. More information here: http://eprints.hud.ac.uk/id/eprint/14051/