How has the Republican Party transformed in values from Abraham Lincoln's era to current times?

In Lincoln's era, the Republican Party was newly formed by anti-slavery Democrats, Free-Soilers, and Whigs, which dissolved as a party in the 1850s. Their main concern at this time was the abolition of slavery. After that was achieved with the Emancipation Proclamation in 1863, the Republicans focused on Reconstruction, but quickly ended this project with their concession to the South in the form of the Compromise of 1877, signed by Republican President Rutherford B. Hayes.
Until the 1930s, most blacks either did not vote (black men did not gain the right until the Fifteenth Amendment was ratified in 1870; black women did not gain the right of suffrage until 1920) or tended to vote Republican, due to the party's association with abolition. Of course, voter intimidation was very common in the South, which was a stronghold for the Democratic Party. This affiliation persisted, even after the sweeping reforms created by the New Deal. Republicans, however, were opposed to Roosevelt's economic plan.
By this time, the Republican Party became the favored party among those in the West, of wealthy people, and of those who lived in the growing suburbs. Most importantly, Republicans were overwhelmingly white Anglo-Saxon Protestants. Meanwhile, urban areas became more heavily Democratic. This shift coincided with the migration of blacks to Northern cities at the beginning of the twentieth century, as well as the influx of immigrants from Southern and Eastern Europe and Asia. These new citizens, particularly blacks, relied on Franklin D. Roosevelt's reforms, even though, arguably, they were not most directly intended for these groups.
The realignment in the Republican Party in the 1930s -- from a party formed to address a social injustice to a party that became increasingly anti-immigrant and staunchly anti-union -- is directly related to Roosevelt's policies and to changes in demographics. The Republican Party's association with white Anglo-Saxon Protestant citizens has persisted to date with few exceptions.
This is not to say that Democrats were more enlightened. During the Civil Rights era, many Southern Democrats, known as "Dixiecrats," were firmly opposed to advancements for blacks. Meanwhile, some Republicans, like former New York governor (and, later, Vice President) Nelson Rockefeller, made great strides against discrimination towards blacks and women in areas such as housing and education. However, Rockefeller was a moderate Republican serving from the mid- to late-1960s, a considerably progressive era, due to Lyndon B. Johnson's leadership. 
Under Johnson, the Civil Rights Act of 1964 was signed, followed by the Voting Rights Act of 1965. Johnson famously said that, with the stroke of his pen, the Democratic Party, to which he belonged, would lose the South forever. His premonition came true. The South, with the exception of West Virginia, became firmly Republican. The spiteful exit of the South from the Democratic Party (though, today, a couple of states have returned and others may soon return) forced a shift in values in the Republican Party. While the party retained its anti-labor platform and its distaste for higher taxes on the wealthy, it also absorbed a more socially conservative strain, drawn from Fundamentalist values that are more intrinsic to the South and parts of the Midwest.
These values became centralized in the Republican Party with the election of Ronald Reagan in 1980. The social values platform on which Republicans claimed to stand remains pertinent to them today. Platforms related to the rights of immigrants, blacks, and women, which began to be addressed in the 1930s, remain more closely associated with the Democrats -- the pro-slavery party of Lincoln's time.