To get back to the issue of post-Civil War parties, it's important to remember that in many respects the Guilded Age parties essentially served as patronage machines, providing government jobs to supporters. So that was an important difference right there, as which party you sided with determined whether you would get tapped for, say, the customs inspector at the local port or not (with its attendant salary).
So part of this distinction lay in where the two parties drew their base of support. The Democrats (outside the South) even then tended to be more associated with the big cities and immigrants (although German Protestants tended to be a backbone of the Republican Party), while the Republicans were heavily associated with big business (especially the railroad industry) and independent farmers (partially a legacy of the Homestead Act, which had been a Republican priority since the beginning, and had been passed only because the Civil War meant that Southern Senators/Representatives were absent), and tended to use the Grand Army of the Republic (the main Union veterans association) as a strong base of support. The Republicans had support from African Americans as well, but the post-Reconstruction era saw most of those voters in the South get disenfranchised as Jim Crow took over, although they still retained support from the smaller Northern black population (and, as a reminder that patronage machines are not all bad, government jobs became an important avenue for economic advancement for African Americans in the Jim Crow period). More broadly, the 19th century Republicans had strong support for the principle of "Free Labor" (which meant anti-slavery, but also "freedom" from things like regulations and unions) and moral improvement (there was a long association between Republicans and the Temperance Movement, for instance, long before Prohibition).
The Democrats spent most of the post-Civil War period in disarray as a consequence of being associated with the Confederacy; on the national level, exactly two Democrats managed to capture the presidency between 1860 and 1932 (Cleveland and Wilson, and note that Wilson benefited from a divided Republican Party in 1912), and were an alliance of separate factions united primarily by their opposition to Republicans and by the desire for patronage, which means their platform tended to be less coherent. A Republican pastor famously characterized the Democrats in 1884 as the party of "Rum, Romanism, and Rebellion," which was unfair but not completely so. Without their solid grip (helped by patronage) in the post-Reconstruction South and in various big cities, it's entirely possible the Democrats might have ended up going the way of the Whigs, but as it stands, they managed to survive and eventually reinvent themselves.
That said, there were very real issues that were debated at the time. Initially you had Reconstruction (which saw a division between Radical Republicans, more moderate Republicans, and Democrats). You also had the issue of inflation; specifically whether to return to the gold standard or allow easier money. During the Civil War, the Union had introduced paper currency not backed by gold ("Greenbacks") to help fund the conflict. This led to minor inflation, which tended to help borrowers (as inflation makes it easier to pay off old loans), but hurt lenders (as their loans issued several years ago were less valuable when paid in current, inflated money). Needless to say, once the war was over, the more connected bankers favored getting rid of the Greenbacks, while many farmers (who tended to be in debt) wanted to promote an inflationary policy. The anti-Greenback folks (who dominated the Republican Party, which was tightly connected with big business interests, as noted above) won that round and enforced a tight money policy, even as the economy collapsed. The inflationists didn't go away however; with Greenbacks gone, they instead wanted to introduce the Silver Standard (or rather, bimetallism) as an alternative to gold. The idea was that rather than your currency being limited by the amount of gold reserves the government possessed, you'd instead have it limited by the sum of your gold and silver reserves, so you could have more money (and thus inflation). This tied in with various other Populists approaches, and became influential (William Jennings Bryan famously captured the 1896 Democratic nomination with a pro-silver platform and a speech proclaiming that "You shall not crucify mankind upon a cross of gold").
More broadly, the late 19th century was dominated by a series of economic crises (Panics) which led to a lot of economic uncertainty and various radical ideas floating around as to how to solve those problems. In addition to inflationists, you had labor unions heating up again, various reform movements, groups forming cooperatives for self-help (such as the various Granges that groups of farmers set up to help each other out), and the like. The degree to which these trickled up to affect the two parties varied, and as noted the Republicans remained dominant on the national level.
Then comes the Progressive Era, which screws around with everyone's ideological alignments.
