During the time right after Reconstruction, there was very little in the way of substantive differences between the parties. They basically agreed on important issues of the day like tariffs and civil service reforms. Instead, what shaped party politics was largely ethnicity and culture. The Republicans were more of a party of "native Americans" and Protestant religions that were descended from Puritanism. The Democrats were more of the party of recent immigrants (and the South) who were Lutheran or Catholic.
Later on in the 1800s, the parties split more, with Republicans being more of the party of the rich and the East, which was why they wanted the gold standard. The Democrats were more of the party of the little guy, which was why they were for "free silver" and why they incorporated the ideas of the Populists.