When did the ideologies of the Republicans and Democrats seemingly switch?

Historically, the Democrats evolved out of the Democratic-Republican (or "Anti-Federalist") Party of the early US. They indeed were usually the party of the South, the party of tradition, the party of planters, and the party of states' rights (against increasing central authority). The figures you would most associate with the party would be Thomas Jefferson - a famous skeptic of state power and a believer in the common man - and, later, Andrew Jackson - who was a believer in mass democracy (at the time, enfranchising more voters - all white men - as opposed to only wealthy / landed men).

Eventually the "Republican" got dropped off the name and they just started calling themselves the Democrats (you can tell how the name would get more than a little long-winded to say...). The party also went through a sort of "rebranding" under Jackson who remade the party more in line with his values in the early 1800s.

Opposed to Jackson were the Whigs (led by Henry Clay) which were more focused on the North. They tended to favor industrialization, entrepreneurship, and merchant interests. They were not especially successful at getting an opponent to the Democrats elected (the Democrats were effectively the ruling party for many of these decades), but John Quincy Adams, a Northerner, did get elected thanks to their support.

Ultimately it was the slavery issue that galvanized support in the North around a new party, the Republican Party, which shared many values with the Whigs and was also openly abolitionist. The early candidates the Republicans ran were the popular captain John Fremont and later Abraham Lincoln.

The election of Lincoln was seen as extremely threatening to the South - which caused secession and civil war. After Reconstruction, the Republicans tended to dominate US politics for the subsequent decades (until new Democrats like Woodrow Wilson and FDR ran in the 20th century). After FDR, there was a very powerful Democratic coalition which was capitalistic, but pro-labor and interventionist, which led us through WWII and dominated our politics for some time.

The shift you are referring to was a result of the JFK/LBJ administration passing the civil rights legislation. This deeply alienated Southern voters at the time. In the subsequent election, Nixon appealed to those disenfranchised voters in the "Southern strategy" which has led to the modern party re-alignment (the North becoming Democratic and the South Republican) that we see today.