At the outset, I will mention that the standard work on this era of history, known as Reconstruction, is the aptly-titled Reconstruction by Eric Foner. If you have any interest in the subject, it will be a rewarding read. It's quite good.

The return of the former States of the Confederacy was, as you might imagine, a contentious issue. There is quite a bit of background to this. In short: Abraham Lincoln had favored a relatively lenient set of conditions for the return of Southern states and their political elites. He constantly battled with the Radical Republicans who controlled Congress from 1864 onward about how exactly Southern states should be readmitted. As an example, the Radicals wanted what was called the "ironclad oath," affirming that an individual had never supported the Confederacy. Lincoln preferred a simpler renunciation of the Confederacy.

With his assassination and the elevation of Andrew Johnson to the presidency, this dynamic remained largely unchanged. Johnson was from Tennessee, and was--if anything--less inclined to be harsh on the Southern states than even Lincoln. Tennessee, which was only incompletely a member of the Confederacy, was admitted under his relatively lenient terms in 1866. After that, the Radicals achieved two-thirds majorities in both houses of Congress, giving them the power to override Johnson's vetoes of their legislation. They used this power to pass the Reconstruction Acts, four laws that set out the terms for the return of the remaining states. The primary purpose of these acts was to install military governments in the Southern states, in order to compel them to ensure suffrage for Freedmen, as well as to write new state constitutions that explicitly banned slavery and guaranteed universal manhood suffrage. The passing of these constitutions was the main term for re-admittance to the Union, and thus the return of their congressmen. They were also required to ratify the Fourteenth Amendment. All of the states, under military government, fulfilled these conditions between 1868 and 1870, with Georgia the last to fulfill them.

As for the atmosphere in Congress, I believe it may have been less tense than you suspect. In most cases, the new Representatives and Senators were elected under these new, Republican-friendly state governments. Many were Republicans, and some were even African American! It was only in 1876, with the contested Hayes-Tilden election and the Compromise of 1877, which allowed the Republican Hayes to become President in exchange for ending military reconstruction. In the following decades the newly-empowered Southern Democrats would replace the Reconstruction-era constitutions with more explicitly racist ones, inaugurating the period we now know of as Jim Crow.