This is one of the few movements in Irish history that was not sectarian in nature; it was the result of the republicanism fostered by both the American and French revolutions. Also, this was the culmination of many years worth of tension/planning against the British. For an overview, please consult this BBC site as well as this Britannica entry. Some recent writings from Trinity College, Dublin should also be perused.
Primary source material can be found in The Rebellion of 1798; hundreds of volumes can be found here (British Library) and here (HathiTrust). The volumes run the gamut from biographies of William Orr, a prominent United Irishman and Theobald Wolfe Tone, a major player and for some, a true martyr for Ireland, to contemporary reports from both sides of the conflict, to works dealing with the French army’s inadequate support of the Irish uprising. Additionally, the National Library of Ireland houses important digital, contemporary resources.
The following government documents MUST be consulted: Report from the Committee of Secrecy of the House of Lords of Ireland, appointed to take into consideration the matters of the sealed-up treasonable papers received from the Commons, on the 23d day of July last: with all the appendices (1798); The speech of the Right Honourable John, Earl of Clare, Lord High Chancellor of Ireland : in the House of Lords of Ireland, Monday, February 19, 1798; and Report from the Committee of Secrecy, of the House of Commons in Ireland, as reported by the Right Honourable Lord Viscount Castlereagh, August 21, 1798 as they all contain valuable primary sources in the appendices of each volume.
The Dublin Castle Collection, the former reference library of the Chief Secretary’s Office in Dublin Castle, contains hundreds of primary sources; for example, here are the results for United Irishmen.
There were numerous pamphlets written during this tumultuous time here are some of them.
There are caricatures from 1798 as well as prints dealing with 1798.
Abstracts of letters related to 1798 are housed in the National Archives of Ireland
The Anti-Union, the very title gives away its intent. It published few issues, but it clearly spoke to the matters at hand.
One can also follow the rebellion through the pages of the London Gazette. Here are some examples: from June 2, 1798; June 5, 1798; June 11, 1798.
The American press devoted much space to this rebellion, reprinting material from the London Gazette and including other accounts as well.
Eyewitness accounts not found elsewhere: Letter from Jane Adams, containing a private narrative of the rebellion of 1798; Aunt Flora Remembers the Irish Rebellion of 1798; and The Irish Rebellion of 1798: A Personal Narrative.
Consult the Dictionary of Irish Biography for numerous entries on United Irishmen.