Were there any early attempts or debates to abolish slavery in the USA, before Lincoln?
Yes, there are quite a few.
The very first was in 1688, when Quakers in Germantown, Pennsylvania wrote a two-page condemnation of the practice and sent it to the governing bodies of their Quaker church.
The Society for the Relief of Free Negroes Unlawfully Held in Bondage was the first American abolition society; it was formed in 1775, primarily by Quakers in Philadelphia. Rhode Island Quakers, associated with Moses Brown, were among the first in America to free slaves.
Thomas Paine wrote one of the first articles advocating the emancipation of slaves and the abolition of slavery in 1775, titled "African Slavery in America".
The US constitution stated that no amendment regarding slavery or direct taxes could be permitted until 1808. This was mostly to give the states time to decide what to do about the matter before an amendment to the Constitution was made.
One of the first to attempt to abolish the slavery trade in the American colonies was Thomas Jefferson. Jefferson included strong anti-slavery trade language in the original draft of the Declaration of Independence, but other delegates removed it. As President, Jefferson signed the Act Prohibiting Importation of Slaves on March 2, 1807, which took effect in 1808 (which was the earliest it could have came into effect). However, whether or not Jefferson was a true abolitionist is debatable, as Jefferson kept hundreds of slaves himself. He privately struggled over the issue of slavery. See this Wikipedia page for more details concerning this issue.
Through the Northwest Ordinance of 1787, the Congress of the Confederation prohibited slavery in the territories northwest of the Ohio River. By 1804, abolitionists succeeded in passing legislation in most states north of the Ohio River and the Mason-Dixon Line that would eventually emancipate the slaves.
One notable person was Robert Carter III of Virginia, who freed more than 450 slaves by "Deed of Gift", filed in 1791. This was more slaves than any other single American had freed or would ever free.
Here is an excerpt from Wikipedia regarding the founding of Liberia:
During the 1820s and 1830s the American Colonization Society (A.C.S.) was the primary vehicle for proposals to return black Americans to freedom in Africa. It had broad support nationwide among white people, including prominent leaders such as Abraham Lincoln, Henry Clay and James Monroe, who saw this as preferable to emancipation. . . . There was however, considerable opposition among African Americans, many of whom did not see colonization as a viable or acceptable solution to their daunting problems in the United States. After a series of attempts to plant small settlements on the coast of West Africa, the A.C.S. established the colony of Liberia in 1821–22. Over the next four decades, it assisted thousands of former slaves and free black people to move there from the United States.
William Lloyd Garrison led a radical shift in the 1830s; he demanded slave-owners to repent immediately, and set up a system of emancipation.
A very important part that cannot be left out is Uncle Tom's Cabin by Harriet Beecher Stowe. Outraged by the Fugitive Slave Law of 1850, Stowe emphasized the horrors that abolitionists had long claimed about slavery.
Nat Turner led the most successful slave rebellion in U.S. history in 1831. The rebellion was suppressed, but only after many deaths.
Isabella Baumfree, a former slave, changed her name to Sojourner Truth and began preaching for the abolition of slavery in 1843.
John Brown led a famous raid in 1859, and seized federal Harpers Ferry Armory, which contained tens of thousands of weapons. Brown believed that the South was on the verge of a gigantic slave uprising and that one spark would set it off. However, he was eventually hung in 1859. He was a major cause of the Civil War.
I've tried to cover the major attempts here, but there were many other attempts that I did not list.
The Transformation of American Abolitionism: Fighting Slavery in the Early Republic
The First Emancipator: Slavery, Religion and the Quiet Revolution of Robert Carter
The Impending Crisis: 1848–1861
The English Model:
The problem with outlawing slavery (or really anything nationwide) in the USA before the Civil War, is that before the war it wasn't really The United States of America, but rather The United States of America. The central government was not all that strong, and was operating under a constitution that made outlawing anything nationwide very difficult. Some have even argued the Constitution was specifically designed to make it nearly impossible to outlaw slavery against the will of the slave states.
The entire backbone of this design was that free states would have to get a 3/4ths majority of states to outlaw slavery. What freaked the South out enough to cause the Civil war was that the Republican party was dead set against expanding slavery to any of the new states. With them in control of the government when it was set to add a whole bunch of new states, this balance was in peril.
So as far as Federal action to outlaw slavery before the war, there really wasn't one. Unless you consider the soft approach of limiting new slave states to be an attack on slavery (which is exactly how the slave states saw it).
That being said, individual states can, and did, outlaw slavery. For instance New York passed an "abolition" law in 1799 that took about a generation to come into effect. So athough it was technically outlawed, there were still legally some slaves in New York at the start of the war.
The Haitian Model:
There were lots of little uprisings that aren't very well known. The two most serious are rather famous though. First was Nat Turner's Rebellion, which was essentially a slave conspiracy and uprising on the Haitian model.
The second, John Brown's attack on Harper's Ferry, wasn't as bloody, but was even more important. That's because it was led by a white man, with the intent of freeing the entire South by raising the slaves in the countryside, much like how Garabaldi united Italy a year later. The raid was easily put down, but John Brown became a hero in the North, and scared sotherners reacted by forming militias that later became the core of the Confederate army. This incident indirectly inspired The Battle Hymn of the Republic.
What actually happened
What actually happened was a combination of both. Contrary to popular belief, Lincoln's Emancipation Proclamation, did not outlaw slavery in the United States. It only freed slaves in rebelling states (which didn't recognize Lincoln's authority anyway). It was remarked at the time that it didn't free a single slave.
However, after the war the passage of the 13th Amendment to the Constitution did outlaw all slavery. The Civil War was a necessary precondition of its passage. The 3/4ths majority was assured by the fact that the old Confedarate states did not yet have full voting rights restored to them. When given a chance, they still voted against it.
Perhaps the earliest attempt to free Negro slaves (at least within their jurisdiction) was by the Quakers of Pennsylvania, late in the eighteenth century.
"Abolitionist" reformers picked up the pace early in the nineteenth century. Perhaps the most famous of them was William Lloyd Garrison, whose first issue of the Liberator in 1831 opened, "I will be as harsh as truth, and as uncompromising as justice. On this subject, I do not wish to think, or speak, or write, with moderation.... I am in earnest—I will not excuse—I will not retreat a single inch—AND I WILL BE HEARD."
Another famous abolitionist in the 1840s and 1850s was the preacher Henry Ward Beecher, and his even more famous sister, Harriet Beecher Stowe, author of "Uncle Tom's Cabin."
Lincoln's election was the culmination of a situation where abolitionists like the ones mentioned above had won the support of perhaps half the country.