The most important thing to remember concerning Lincoln's views on racial equality and race is that they were at some points consistent with what one would expect to find in the nineteenth century, but were also by no means rigid—particularly towards the end of his life, they had changed immensely compared to what they were prior to his election and the early part of the ACW. I'll divide my answers into a few distinct issues that all tie into race:
Slavery: Lincoln's first public statements against slavery can be found as early as 1837, while the correspondence we have as well as the collection of speeches throughout his lifetime reveal a very powerful resentment of slavery due to moral conviction. Just as an introduction to his words on the matter, the NPS has a good website devoted to this, which shows various statements he made on slavery chronologically. One thing to keep in mind here is that Lincoln, while anti-slavery, ascribed to an ideology that was distinct from abolitionism in one crucial aspect: anti-slavery in the antebellum period refers to those who wanted to halt the expansion of slavery, and ideally see its eventual death. The Republican party was essentially founded on this position, absorbing a good deal of the old whigs that held similar views. Abolitionism, on the other hand, was viewed as a radical (sometimes dangerous, depending on who you asked) minority that wanted to see the immediate death of slavery through government prohibition of the institution itself, as well as those institutions necessarily connected to it (like the interstate slave trade and the body of laws that indirectly enabled the practice). Anti-slavery advocates like Lincoln, however, generally believed that the federal government had no right or duty to interfere with the South's "Peculiar Institution" (slavery), it being something implicitly protected by the Constitution, the laws of those states, and certain federal policies as well. Even through the secession of the Southern states and the onset of the war, Lincoln made it very explicit that he had no intention of interfering in Southern slavery. He was a devout unionist, much like his favorite statesman Henry Clay, and this devotion came before others. However, as the war begins to drag on, it becomes clear and clearer that slavery is so tied up in the war that dealing with this source of division has to be done, in some form or another. (Lincoln was also motivated by a belief that the war was in a sense a divine punishment for the practice of slavery, and viewed emancipation and later abolition as both practical moves to solve the source of disunion, as well as a kind of redemptive action. He indicates this, among other places, in the Gettysburg Address as well as his Second Inaugural Address.) As early as spring 1860, possibly earlier, it becomes clear that Lincoln is starting to come around to the idea of emancipation (with heavy pleading from the abolitionists he allowed make him their audience), and by late summer/fall of 1862 he's waiting to issue his preliminary Emancipation Proclamation, which frees slaves as a war measure in Union-controlled territory (excepting border states and certain counties and parishes in the South under Union control).^1 It's not until the end of the war that Lincoln begins to push abolition by Constitutional amendment, which passes through Congress in late 1864/early 1865, prior to the end of the war.
Legal equality of races: this one's a bit murkier, as Lincoln did believe that the Constitution and the spirit of the founding itself did hold that all men are created equal, and ought to be afforded the same basic rights. This did not extend, at least a slight intimation contrary just before his death, to full political equality (e.g. suffrage and ability to serve on juries for certain educated black persons). Mind you, this is only just after the passage of the Thirteenth Amendment, and was the first time that any president had ever so much as even mentioned the extension of such rights to black persons, much less suggested it as policy. Relatedly, earlier in his life you do see certain instances of Lincoln, as a practitioner of law, defending the property rights of slaveowners over their slaves, which is a bit unexpected given the lionizing of Lincoln that tends to happen in basic historical education in the U.S. Of course, this does not suggest that Lincoln agreed with the practice of slavery in any way, except as an unfortunate consequence of the laws as they were written and as they stood.
Equality of races themselves: There are no doubt instances of Lincoln expressing racist viewpoints, as you would expect from a white man living in the nineteenth century.^2 You see such opinions expressed in 1858 while campaigning against Stephen Douglas, as well as on other occasions. (Zinn in A People's History loved mixing this into the corresponding chapter.) These tend to contrast with other statements from Lincoln, such as expressions of embarrassment with the practice of slavery in the U.S. despite the ideal projection of American Republicanism in his view, ^3 as well as expressions of genuine sympathy for the conditions of slaves such as the one in the letter to his friend Jonathan Speed sourced above. Another key area to examine is Lincoln's stance on colonization, which stemmed in part from his reservations concerning the place of black individuals in American society. A fair case can be made that Lincoln's devotion to this policy was not and expression of blatant racism in and of itself (or at least anything unique to Lincoln), but rather symptomatic of the society where both northerners and southerners did not want to live alongside blacks, except under the condition that they be subject to legally enforced inequality. It's disputed as to when Lincoln truly abandoned this position. Around 1862, in one of his messages to Congress, he did express distaste for those who had a genuinely malicious intent for their favoring of colonization, and it's not hard to make the case that Lincoln began to disfavor colonization altogether when abolition and the possibility of a post-war revolution in the South became more feasible.
Sources:
Pretty much any biography of Lincoln, such as those by elsewhere-cited Eric Foner, as well as Doris Kearns Goodwin's Team of Rivals (in spite of its tendency to sensationalize and leave out certain themes), or Stephen Oates' With Malice Towards None. Various works on the antebellum period or the ACW will also contain a good deal of this information, such as McPherson's Battle Cry of Freedom, or David Potter's The Impending Crisis.
Anyway, I hate to say that my response is a rather bare outline of Lincoln's positions, so I hope one of the antebellum/Civil War folks can add anything I've left out.
Notes:
There is a lot of piffle that you find throughout on the Emancipation Proclamation, much of it condemning Lincoln for it, while some misrepresents its intent and scope. As an accessible source, this interview of Eric Foner and the Brooklyn Historical Society is a good watch for getting more information on that document and its context.
I myself am not quite sure of the extent to which Lincoln held these beliefs, expressed them due to them being the only truly viable statements on race for any politician in the period, or both—if such a thing is generally known, in which case I would defer to anyone else who does know and can source this.
To provide an example, I'll take a quote form the Lincoln-Douglas Debates (1858): "This declared indifference, but, as I must think, covert real zeal for the spread of slavery, I cannot but hate. I hate it because of the monstrous injustice of slavery itself. I hate it because it deprives our republican example of its just influence in the world—enables the enemies of free institutions, with plausibility, to taunt us as hypocrites—causes the real friends of freedom to doubt our sincerity, and especially because it forces so many really good men amongst ourselves into an open war with the very fundamental principles of civil liberty—criticizing the Declaration of Independence, and insisting that there is no right principle of action but self-interest."