I read about how the Democratic party supported segregation and slavery and the Republican party opposed them. How did the Democratic party change and become the party to put the first black president in the White House?

I've taken a long hiatus from intellectual Reddit, but appreciate the callback! Time to wade back in:

/u/Okleconservative, I really can't recommend strongly enough the sources flagged in my linked answer (also listed below).

But I warn you, they're going to challenge the premise of your question. It's not quite fair to say the Democrats "supported segregation and slavery" -- unless you go back before 1900 -- because the parties weren't anywhere near as ideologically rigid in the civil rights era, and the politics of race were excruciatingly complicated. To build his civil rights-reform coalition, for example, Lyndon Johnson had to meld bicoastal "Rockefeller Republicans" with progressive Democrats, while placating segregationist Democrats, like his mentor Richard B. Russell. That means it's also not quite fair to say Republicans "opposed" Democrats on civil rights.

None of that fits neatly into a talking point, so you'll see this history oversimplified (by both parties) quite frequently! Some good books for further reading, and what part of your question each answers:

For background & 1940s-era Southern politics: Robert Caro, Path to Power: background and vignettes on Southern Democrats, progressive and otherwise, like Sam Rayburn, & Roosevelt's popularity among them.
For 1950-1960: Robert Caro, Master of the Senate: How Johnson, as Senator, managed to pass the first civil rights bill since Reconstruction.
For 1960-1964: Robert Caro, Passage of Power: how Johnson adopted Kennedy's civil rights agenda as his own. (The "why" question is answered in Path to Power.)
For 1964-present: Earl and Merle Black, The Rise of Southern Republicans: how that legacy, combined with Nixon & Reagan, "realigned" the South in to Republican hands.