Eric Foner's The Fiery Trial talks a fair amount about this. Lincoln never actually came out in support of full equality as we would understand it, and probably always sincerely opposed interracial marriage, for example. In his famous debates with Stephen Douglas in 1858, Lincoln argued that a black person "is not my equal in many respects—certainly not in color, perhaps not in moral or intellectual endowment. But in the right to eat the bread, without the leave of anybody else, which his own hand earns, he is my equal and the equal of Judge Douglas, and the equal of every living man."

However, Lincoln's views on many issues, such as voting rights for blacks, did gradually change over time, almost always in the direction of greater equality. Foner suggests that a combination of personal meetings with prominent African-Americans such as Frederick Douglass, the role of black soldiers in the Union army, and shifts in public opinion led him to change his views. If Lincoln had lived longer, would he have supported the 14th and 15th amendments? It is perhaps dangerous to engage in counterfactuals of this sort, but some leading Radical Republicans at the time felt that he was leaning toward their views when he died.

For a long time before his presidency, Lincoln was not at all an abolitionist as we often think of him. Though morally opposed to the existence of slavery, he had complicated views on the subject of the rights of African Americans, as he also recognized the rights of slave owners to some extent. During the famous Lincoln-Douglas debates in 1858, he explicitly called the white race "superior"; he also came out against miscegenation and the suggestion of blacks serving on juries. Further, during his presidency Lincoln repeatedly stated that he would save the Union even if it meant permitting the continuation of slavery in the South. See for example this August 22, 1862 letter to Horace Greeley. It was not until the war progressed that he began to support manumission, with the Emancipation Proclamation and his advocacy for the 13th Amendment representing the finality of this evolution. Why exactly this happened and what this meant for his social equality beliefs is still up to debate, though he did support giving freed slaves citizenship (which is actually the issue that may have most inspired John Wilkes Booth, who was present when that speech was given). It's a complicated subject, one that Lincoln biographers have been struggling with for years.

I guess basically what I mean, is how progressive were his views? Would he have any issue with something like interracial marriage?

This is a difficult question to answer because it asks us to speculate, but I am sure that the answer is neither black nor white. Like most political figures in history, Lincoln was bound by pragmatism and realpolitik. Would Lincoln have wanted complete equality for Blacks and Whites? It is hard to say, the only person who would have known that would be Lincoln himself or perhaps his closest confidants. What we do know is this: Lincoln accepted that the will of the nation did not allow for perfect equality. Frederick Douglass probably summarizes this political reality best, saying this of him (post mortem): "[Whites] are the children of Abraham Lincoln. We are at best only his step-children; children by adoption, children by forces of circumstances and necessity."

Here is a longer excerpt from the oration:

We fully comprehend the relation of Abraham Lincoln both to ourselves and to the white people of the United States. Truth is proper and beautiful at all times and in all places, and it is never more proper and beautiful in any case than when speaking of a great public man whose example is likely to be commended for honor and imitation long after his departure to the solemn shades, the silent continents of eternity. It must be admitted, truth compels me to admit, even here in the presence of the monument we have erected to his memory, Abraham Lincoln was not, in the fullest sense of the word, either our man or our model. In his interests, in his associations, in his habits of thought, and in his prejudices, he was a white man.

He was preeminently the white man’s President, entirely devoted to the welfare of white men. He was ready and willing at any time during the first years of his administration to deny, postpone, and sacrifice the rights of humanity in the colored people to promote the welfare of the white people of this country. In all his education and feeling he was an American of the Americans. He came into the Presidential chair upon one principle alone, namely, opposition to the extension of slavery. His arguments in furtherance of this policy had their motive and mainspring in his patriotic devotion to the interests of his own race. To protect, defend, and perpetuate slavery in the states where it existed Abraham Lincoln was not less ready than any other President to draw the sword of the nation. He was ready to execute all the supposed guarantees of the United States Constitution in favor of the slave system anywhere inside the slave states. He was willing to pursue, recapture, and send back the fugitive slave to his master, and to suppress a slave rising for liberty, though his guilty master were already in arms against the Government. The race to which we belong were not the special objects of his consideration. Knowing this, I concede to you, my white fellow-citizens, a pre-eminence in this worship at once full and supreme. First, midst, and last, you and yours were the objects of his deepest affection and his most earnest solicitude. You are the children of Abraham Lincoln. We are at best only his step-children; children by adoption, children by forces of circumstances and necessity.

Frederick Douglass, Oration in Memory of Abraham Lincoln 1876

While we can't know what Lincoln was thinking, we do know what Reconstruction Republicans were thinking and it's hard to believe that Lincoln would have opposed them to the extent President Andrew Johnson did during his term and it's hard to imagine history changing (i.e. Jim Crow not happening) if Lincoln had not been murdered.

The 14th amendment and 15th amendments arose out of two main concerns: Dred Scott was still the land of the law at the time of the end of the civil war and the ruling suggested blacks had no citizenship rights to speak of. And second, the 3/5th compromise was still a part of the constitution. The worry was that Southern Democrats would deny citizenship to freed slaves but still use the freed population's numbers to bolster their representation in congress while denying freedmen the ability to cast a ballot. Not an unfounded fear.

Republicans took this stance in part due to the often violent response to black civil rights in the South in the period after the Civil War. Congress passed a series of acts, known as the Reconstruction acts to break the back of the Klu Klux Klan and attempt to guarantee social equality to freed blacks living in the South.

The first attempt to rectify the situation was the Civil Rights Act of 1866, which granted blacks full citizenship and overturned Dred Scott. But many thought that was it was inadequate and would be overturned by the Supreme Court. That was the genesis of the 14th amendment, which sought to ensure that Dred Scott was overturned not just in law, but in the constitution. The 15th amendment was seen as vital not just for the rights of freed blacks, but for the future of the Republican party, since most saw that if blacks were granted the right to vote, the Republicans could be competitive in the states of the former confederacy. If black voters were stricken from the voting rolls, Democrats would have a solid lock on the South for generations to come (which is obviously what happened).

Another point to make is the distinction between civil rights and social rights. Today we tend to see them as one in the same (the result of advocacy by many individuals and groups over the last century) but back then social rights were a thing apart from civil rights. While many people believed blacks had the right to vote, fewer people believed that blacks deserved social equality. Even so, Radical Republicans passed the civil rights act of 1870 in response to the black codes in the South to guarantee that blacks could not be discriminated against in jobs, public accommodations, etc. The law is very similar to the act passed by President Lyndon Johnson a century later.

What happened was that the Supreme Court eventually ruled the Civil Rights Act of 1870 unconstitutional and gutted the meaning of the 14th amendment in the Civil Rights cases. In the civil rights cases, the Court ruled that Congress lacked the power to ban discrimination by private individuals rather than state and local governments. Later rulings, most notably Plessy v. Ferguson, would state that segregation by state and local governments did not violate the social equality of blacks. Later cases would validate strategies by Southern Democrats to prevent blacks from voting, since at the time decided eligibility to vote was seen as a state power, not a federal power so long as the law didn't explicitly ban voting on the basis of race. By then, the threat from the Klu Klux Klan had been mostly eliminated and eventually Southern Democrats gained enough power in congress to block any further civil rights action. This culminated in the election of 1876 which in exchange for Rutherford B. Hayes assuming the presidency, Republicans agreed to withdraw the military from the South, which allowed Jim Crow to flourish.

It's hard to imagine this history changing if Lincoln is not murdered. There are too many moving parts, too many individuals involved for one man to have changed this outcome.

A good book that discusses the history of the 14th amendment during reconstruction is No Easy Walk to Freedom: Reconstruction and the Ratification of the Fourteenth Amendment by James Bond.