I'm going to answer this more about nationalism in general, and not specifically independence, for reasons that I hope will become clear over the course of the answer.

The first major, sustained campaign for "independence" could be considered Daniel O'Connell's (1775-1847) campaigns for the Repeal of the Act of Union, which came into effect in 1801. The Act of Union ended the Kingdom of Ireland that had existed since 1542, and with it the independent Irish Parliament in Dublin. However, while that Kingdom was ostensibly an independent country, it was in effect an extension of British authority, with the Irish Parliament's acts needing to be approved in London.

O'Connell's genius lay in building a mass movement amongst ordinary people. He rose to prominence by winning election to the House of Commons in 1828 despite being unable to take his seat as a Catholic. In the 1820s and 30s, through his Catholic Association, O'Connell represented the needs and ideals of Irish Catholics against the structural disabilities and inequalities in Ireland. These included payment of tithes to the Protestant Church of Ireland, the state church, and resistance to the practices of landlords and the inequality of landholding, as well as political exclusion. Crucially, when he launched his campaigns for Repeal in the 1840s, based on these existing campaigns, he did several important things: 1. He spoke in English, even though he also spoke the Irish language. 2. He tied the political-constitutional settlement of Repeal to an economic issue of landlordism and the massive disparities of landownership, where most Irish land was owned by a small number of elite Anglican Protestant landlords (in the 1861 census, they were 12 percent of the Irish population but owned over 90 percent of the land). 3. He never actually defined what Repeal meant, or what his proposed settlement actually was, allowing for a misty, romanticised ideal of what independence or self-government would actually look like. Repeal could be all things to all people.

O'Connell's formula of popular political organising as the muscle behind a conflation of economic, Catholic-cum-social, and constitutional issues never really went away. It became the template for many other movements.

There's an important contrast between the rural, Catholic working class represented by O'Connell and the urban, industrial working class emerging in Ulster, in the North of Ireland. Already tied closely to Scotland due to migration patterns in previous centuries, Ulster was already industrialized: shipbuilding in Belfast, linen elsewhere. It was part of an emerging industrial triangle of Belfast-Liverpool-Glasgow. Famously, the Presbyterian minister Henry Cooke referenced the industry and wealth gained through the British connection and said "Look at Belfast and be a Repealer if you can."

The Famine (1845-51) massively exacerbated existing tensions, to put it mildly. Roughly a million Irish people died and another million emigrated. Most immediately, many blamed landlordism, and especially absentee landlordism, as the source of Ireland's problems. One of the earliest responses by the British government was the Incumbered Estates Act of 1849, which tried to bring new wealth and better landlords into the country (it didn't work). Post-Famine, there were repeated attempts to solve the landlord-tenant question, such as the 1870 Land Act.

Also in 1870, a group of Protestants and Catholics got together in Dublin under the leadership of the Protestant Isaac Butt (1813-1879) to form the Home Government Association, a political club that advocated, like O'Connell, for some form of Irish self-government. Why were Protestants involved? That last independent Irish Parliament had been exclusively Anglican, after all. The Kingdom of Ireland was a Confessional Anglican State, despite the overwhelmingly Catholic population of Ireland. Protestants could look back to their grandparents' days of Ascendancy Rule. Catholics in the HGA could look forward to more of the political, social, and economic gains that had been increasing over the century: Catholic Emancipation in 1829, Reform Acts, 1869 Disestablishment of the Church of Ireland, the Land Act, etc. The HGA became the Home Rule League in 1873, to more effectively fight electoral battles, and eventually became a Nationalist political party under the Protestant Charles Stewart Parnell (1846-1891).

Economically, self-government didn't have much appeal for the strongly-protestant Ulster, because industrialization and the value of the British connection had only grown over the century. There is also the undeniable cultural difference that many Ulster protestants were Presbyterians, culturally and educationally linked with Scotland, while protestants in the rest of Ireland were often Anglican, and thus part of a Church that claimed to be the church of St Patrick and the true inheritor of medieval, pre-reformation Irish Christianity. While the Irish Anglican elite were often educated in England and held land in both countries, there was a strong nationally Irish strain within that community, even though it wouldn't have seemed very nationalistic to their Irish Catholic tenants.

The turning point, as I see it, is the 1879-181 Land War. This was a campaign of rent strikes designed to break the power of landlords once and for all. Refusing to pay rent, killing landlords, and waves of evictions followed. The Irish Parliamentary Party, as the Home Rule League had become, supported the Land War, even though the Irish Land League was officially a separate institution. At this point, the O'Connellite conflation of land reform and nationalism truly came together, and there was almost nothing in the nationalist program that appealed to either Northern industrial worker protestants nor Southern landholding protestants.

Both Conservative and Liberal British governments introduced further Land Reform Acts (1881, 1885, 1887, 1903). British Liberal governments under W.E. Gladstone (1809-1898) introduced a series of Home Rule bills in 1886 and 1893, which were vociferously fought by protestant Ulster. This culminated in crisis of 1912-14, where the civil unrest the prospect of Home Rule brought was profound.

By the First World War, the generation of so-called Constitutional Nationalists who'd been willing to work with and through Parliament to achieve Home Rule were old has-beens, and in 1916 the Easter Rising completely changed the landscape. Either it was the spark that began a movement for independence long delayed, or it was a stab-in-the-back betrayal during the bloodbath of the Somme. The British Government's harsh reprisals to 1916 further radicalized a younger generation into open warfare in which clear Protestant/Catholic lines had been drawn. As Ireland descended into warfare in 1919-1921, the generations of political organizing earlier ensured that the sectarian divisions were clear.

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I can't give you a bibliography of all of Irish history in the C19th, but some selected sources are:

Alvin Jackson, Home Rule: An Irish History (Oxford, 2003).

JJ Golden, "The Protestant Influence on the Origins of Irish Home Rule, 1861-1871," The English Historical Review, vol. 128, No. 535 (Dec., 2013) pp. 1483–1516.

C. Nic Dhaibheid, "The Irish National Aid Association and the Radicalization of Public Opinion Ireland, 1916-1918," The Historical Journal, 55(3), 705-729

Richard English, Irish Freedom: The History of Nationalism in Ireland (London, 2006)

Colin Reid, "‘An Experiment in Constructive Unionism’: Isaac Butt, Home Rule and Federalist Political Thought during the 1870s", The English Historical Review, vol.129, no.537(April 2014), Pages 332–361