The effects of acid rain were noticed as early as 1667, when John Evelyn noted the deterioration of the Arundelian marbles:

When I saw these precious monuments miserably neglected, and scattered up and down about the garden, and other parts of Arundel House, and how exceedingly the corrosive air of London impaired them, I procured him to bestow them on the University of Oxford.

in his diary. A little earlier, the acidic corrosivity of coal smoke had been noted by Kenelme Digby, son of a Gunpowder Plot plotter, courtier, pirate, diplomat, and scientist, in his A Discourse on Sympathetic Powder of 1658:

Coal hath .. volatile salt very sharp

(quoted in Brimblecombe (2001)). Effects of acid rain due to volcanic eruptions were scientifically observed in the 17th and 18th centuries, especially due to the very large 1783 Laki eruption in Iceland.

The first clear identification of acid rain as such, and the name "acid rain" came later, both by the Scottish chemist Robert Angus Smith in 1852 (Reed, 2014). The two main culprits of acid rain at the time were sulphur dioxide from coal burning (probably the cause of John Evelyn's acid rain) and hydrogen chloride from soda ash production. The Leblanc process for producing soda ash could be easily made cleaner, since hydrogen chloride is very soluble in water (producing hydrochloric acid), and emission reduction was enforced by law under the British Alkali Act of 1863. Smith was given the post of Chief Inspector of Alkali Works, and this legislation successfully limited the growth of hydrogen chloride emission (the reduction in emission per plant combined with an increase in the number of plants resulted in approximated constant emissions). In 1872, his Air and Rain: The Beginnings of a Chemical Climatology which discusses acid rain and its effects was published.

Sulphur dioxide emissions continued to rise, with European emissions rising by a factor of about 8 in the century from 1880 to 1980 when European SO2 emissions peaked (Grennfelt, 2020). Widespread use of the internal combustion engine also contributed nitrogen oxides to the problem; the combined emission of sulpher and nitrogen oxides in Europe in 1980 was about 10 times what it had been a century earlier. It is this large increase in the 20th century than was responsible for the modern attention to acid rain from the 1960s onward, beginning with observation of the acidification of lakes in Scandinavia. This modern attention led to legislation, e.g., the US Clean Air Act Extension of 1970 which set limits to allowable emissions of nitrogen and sulphur oxides. This led to a reduction in such emissions in the West - today, European emissions are less than double what they were in 1880 (Grennfelt, 2020). However, increased emissions from East Asia and SE Asia mean that the global reduction in SO2 emissions has been small, only about 15% from the 1990 global peak, and the increase in Asian car numbers resulting in almost no global decrease in nitrogen oxides (Fowler, 2020).

References:

John Evelyn, The Diary of John Evelyn, 2 vols, M. Walter Dunne, 1901: http://www.gutenberg.org/ebooks/41218 and http://www.gutenberg.org/ebooks/42081

P. Brimblecombe, "Acid Rain 2000±1000", Water, Air and Soil Pollution 130, 25-30 (2001).

Peter Reed, Acid rain and the rise of the environmental chemist in 19th century Britain: the life and work of Robert Angus Smith, Ashgate, 2014.

Robert Angus Smith, Air and Rain: The Beginnings of a Chemical Climatology, Longmans, Green, and Co., 1872: https://archive.org/details/airrainbeginning00smitiala

Grennfelt, P., Engleryd, A., Forsius, M. et al., "Acid rain and air pollution: 50 years of progress in environmental science and policy", Ambio 49, 849-864 (2020). https://doi.org/10.1007/s13280-019-01244-4

Fowler, D. et al., "A chronology of global air quality", Phil. Trans. R. Soc. A 378, 20190314 (2020). http://dx.doi.org/10.1098/rsta.2019.0314 [open access]