The Thirteenth Amendment, because it was passed immediately in the wake of the Civil War, had almost immediate effect in the sense that it banned chattel slavery. However, as Reconstruction collapsed, African-Americans and many poor whites were often driven by necessity and by a white-controlled economy into arrangements that often resembled slavery. Many sharecropper contracts, in addition to establishing onerous economic terms, also gave property owners the ability to confiscate property and even, in some cases, physically punish sharecroppers. So the letter of the Thirteenth Amendment had immediate effect, while its spirit was violated as a matter of course in the South.
It was precisely the existence of "vestiges of slavery," including these contracts and the emergence of so-called "Black Codes" that brought about the Fourteenth Amendment. It was designed to guarantee equal protection under law to all American citizens, which under its definition, included virtually all living slaves. But extremely limited readings of the Fourteenth Amendment by the Supreme Court in a series of decisions in the late nineteenth century opened a legal space for states to deny the protections of the amendment to African Americans. The "separate but equal" doctrine outlined in Plessy v. Ferguson (1896) gave legal sanction to racial discrimination by state governments.
The Fifteenth Amendment was an acknowledgement of the reality that African-Americans could never achieve legal equality in the South without political power. It thus extended the franchise to all men regardless of race or condition of prior servitude. Yet, as is well known, southern governments found loopholes to avoid granting the franchise, and after Reconstruction collapsed in 1876, blacks were denied the vote both through legal devices such as poll taxes, literacy tests, and even selective gerrymandering, and through the use of terror and other repressive measures. The full benefits of the Fifteenth Amendment were not realized until the Voting Rights Act of 1965, the result of a long civil rights struggle.