Short answers, pulled from the story.
The Slavery Abolition Act received royal assent on the 28th of August 1833. All slaves in the British Empire except those in India were emancipated on the 1st of August 1834, though a transitional apprenticeship system delayed full legal freedom until the 1st of August 1838.
The Slave Trade Act was passed by the British Parliament on the 25th of March 1807, making the slave trade illegal throughout the British Empire. It imposed a fine of £100 for every slave found aboard a British ship and was partly enforced by the Royal Navy's West Africa Squadron.
James Somersett was an enslaved African whose 1772 court case in England forced Lord Mansfield, Chief Justice of the Court of the King's Bench, to rule that slavery had no basis in English common law. His judgment of the 22nd of June 1772 led to between 10,000 and 15,000 enslaved people gaining freedom in England by 1774 and helped launch the wider abolition movement.
Parliament set aside £20 million to compensate slave owners for the loss of their enslaved people under the 1833 Act. This sum can be compared with the UK's total government spending of £51.5 million and a GDP of £443 million in 1832. Formerly enslaved people received no compensation or reparations.
Thomas Clarkson was the abolitionist movement's primary researcher and organiser. He interviewed sailors and former slaves at ports including Bristol, Liverpool, and London to gather firsthand evidence, and built a national network of 1,200 local abolition groups. Petitions he helped organise gathered 400,000 signatures in 1792 and 1,375,000 signatures in 1814.
The British and Foreign Anti-Slavery Society was founded in 1839 to campaign against slavery in other countries after formal emancipation in the British Empire. It is still in operation today under the name Anti-Slavery International, based in London, and is recognised as the world's oldest international human rights organisation.