Were there any early attempts or debates to abolish slavery in the USA, before Lincoln?
Yes, there are quite a few.
I've tried to cover the major attempts here, but there were many other attempts that I did not list.
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.