I'd say that beside anti-slavery, the most salient ideological point of the Republican party was economical protectionnism and federal investment.

Being, at least in part, the heir of the Northern Whig Party, the Republican party inherited its tendancy to favor higher tariffs and a platform of federal investment in infrastructure. This orientation can easily be traced back to the famous American System of Henry Clay, an all-encompassing economic policy articulating tariffs, a National bank and massive federal intervention.

While this aspect took a step back in the mid-XIXth because of how important slavery got in the public space, it became a trademark of the Republican party for the rest of the century, with many political campaigns revolving around the oppositions between Rep and Dem around the tariff question and the Federal vs State/private management of public infrastructures.

So, from that point of view, I'd argue that XIXth century was pretty much the opposite of the modern-day GOP. Even tho, comparisons like this are always tough, because there are many differences in context, and you can't just say "well the modern-day GOP economic policies are just like the Democratic economic policies of the XIXth century !"

The second ideological aspect that I'd highlight would be the nativist inclinations of the mid XIXth century Republican party. Then again, it stems from its roots, with the influence of the Know-Nothing movement and the American Party. Although those wings weren't necessarily dominant in the Republican Party, as the Lincoln nomination kinda demonstrate, they infused Republican ideology for the years to come and reappeared in different forms until late in the XIXth century.

For example, the 1870s Liberal-Republican movement that emerged as a critic of Grant's administration massive corruption was defending a very nativist (and racist) perspective, wanting to restrict suffrage and exclude immigrants from the public arena, denouncing, for example, Irish as responsible for the degradation of the American spirit.

This WASP-ish nativism and anti-immigrationisme appeared again the 1880 : the most famous example is probably the Chinese Exclusion Act of 1882. But we can find more pervasive examples in other aspects. For example, during the presidential campaign of 1884, it's been often said that Blaine lost the election by failing to react to a protestant bishop calling the Democratic party the party of "Rum, romanism and Rebellion".

Whether it actually caused his defeat is beyond the point. The fact is that the accusation of "romanism" as derogatory was pretty usual from the Republican press to criticize non-WASP democratic-voting populations. From that aspect, you could maybe draw a line from this protestant, anti-immigrationist nativism to current-day GOP, especially the brand carried by Trump. But, once again, it's probably more complicated than that.

I'm not a specialist of political ideologies so maybe I missed a major point that someone else will highlight, but as a specialist of political campaigns, those are the two point that sprang to mind for that period.

Sources :

Battle Cry of Freedom: The Civil War Era - James McPhearson

The Republic for Which it Stands - Richard White

I have more precise sources for the 1884 campaign and the Liberal-Republican movement if you want, but that's a bit off-topic.