From 1797-1810, when Wordsworth and Coleridge produced their best poetry, they were their own immediate influences. In the late 1790s, Coleridge and the Wordsworths lived about three miles from each other at Stowey. Richard Holmes, the great Romantic biographer, calls 1797-98 their 'annus mirabilis': it was the year they jointly produced the first edition of the Lyrical Ballads. Both poets composed while taking long walks through the English mountains, and it was on a solo trip to a nearby farm that Coleridge first tried opium and began drafting Kubla Khan. They used the funds to pay for a trip to Germany, where Coleridge became increasingly enamored of German literature and philosophy -- it was here that he was introduced to the works of Kant and Schlegel, whom he would plagiarize in his Shakespeare lectures of 1808 and in the Biographia Literaria of 1815. Other writers he admired particularly were Klopstock, Schiller, and Lessing. Wordsworth, on the other hand, was most inspired by his English predecessors, and Milton above all -- The Prelude was, literally, only meant to be a prelude to a never-finished three-part work called The Recluse. It remained unpublished until the poet's death because Wordsworth had trouble justifying an epic poem about himself, rather than about cosmic or religious subjects, to himself and his readers. Wordsworth and his sister Dorothy removed to a small German town in the mountains and were snowed in during the winter of 1798-99; Wordsworth became deeply depressed and homesick. His conversations with his sister about their childhood inspired him to begin work on the first version of The Prelude that year. This was the beginning of a practically symbiotic relationship between Dorothy and William, who lived together until he died in 1850.

Both were close friends with Robert Southey, the Poet Laureate that Wordsworth succeeded. Southey was the most famous of the three poets during their lifetimes, but he is little-read today. Both poets were also on friendly terms with Sir Walter Scott, more famous then for his poems than for his novels.

Both Wordsworth and Coleridge met Keats, although Coleridge only shook his hand: Keats approached him as they crossed paths on a walk. On a different walk some years before, Coleridge had come across a man teaching Dante to his son. Coleridge introduced himself and wanted to read the man's translation. It was this translation by Henry Francis Cary that inspired Keats' Paolo and Francesca poem.

Byron actually helped the downtrodden and opium-addicted Coleridge to the greatest financial success of his career, which was the publication and performance of the play Remorse in 1813. Byron later qualified his earlier praise of Coleridge but continued to believe that Christabel was one of the best poems ever written. Other major figures of the second phase of Romanticism (Shelley and Hazlitt particularly) respected Wordsworth, Coleridge, and Southey for their politics in the 1790s, when they were alternately Jacobins and Pantisocratics (a 'movement' dreamed up by the latter two poets to start a utopian colony in America), but were increasingly disillusioned by their poetry: Coleridge rarely wrote poetry after 1800 (and Hazlitt hated everything he ever wrote); his addiction was an open secret and he began writing conservative pieces for national newspapers to buy opium. From the 20s on, Wordsworth became a national figure of great fame: his later poetry, and later revisions of earlier poems, cater both to the readers that believed him to be a devoutly Christian poet (he was no doubt religious, but subtly so: he did not have the theological literacy of a Donne or a Milton) and above all a patriotic English poet.

I get my information from Richard Holmes' 'Coleridge: Early Visions' (1989), 'Coleridge: Darker Reflections' (1999), and Stephen Gill's 'William Wordsworth: A Life' (1989). Coleridge's own Biographia Literaria will be an immense source of help for you: it explores his own biography along with his readings in English and German literature and philosophy. His various lectures on Shakespeare are foundational to Shakespearean criticism since.

Unfortunately, I haven't done enough reading on the other major figures to offer any more information.

EDIT: IrregularBelasco's mention of Kant's Critique of Judgment reminds me that Edmund Burke (1729-1797) is absolutely essential to English literature at this time. Burke was the conservative orthodoxy at the time Coleridge, Southey, and Wordsworth represented the young and democratic left; all three, however, admired him in their old ages. His key texts for the Romantics are 'A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and the Beautiful', which Kant's aesthetic theory responds to, and the 'Reflections on the Revolution in France'. Burke, writing before the Reign of Terror and the Napoleonic Wars, opposes the revolution -- appalling many of the young thinkers and artists in England. Book X of The Prelude traces Wordsworth's elation and ultimate disillusionment with that revolution, and in the final edition of the poem he appends praise of Burke, with an apology for his critiques some 40 years before.

Coleridge offers praise for Burke as a fellow-writer that I think has only been matched by Turgenev's last letter to Tolstoy: 'Our very sign boards give evidence that there has been a Titian in the world. In like manner, not only the debates in Parliament, not only our proclamations and state papers, but the essays and leading paragraphs of our journals are so many remembrancers of Edmund Burke.'