What was the main goal for northern states to accomplish during Reconstruction

The main goal was to re-incorporate the rebel states back into the Union. The Civil War was over, and there was a general desire for the United States to move forward as one nation. The question, however, was how to do this. After such a monumental and bloody conflict, it was inevitable that there was still a good deal of animosity in the air. Many in the North felt that the Southern states should be punished. They had, after all, started the war by unilaterally seceding from the Union. Technically, this was an act of treason, and Northern partisans strongly believed that this most serious of crimes should be punished accordingly.
On the other hand, there were those, like President Lincoln, who favored a more gradual, conciliatory approach to Reconstruction. For the likes of Lincoln, little was be to gained in re-fighting the war on the political front; the battle-scarred country needed to move on, both North and South together. It would still be necessary for the federal government to impose political changes on the Southern states; but this would be carried out in a spirit of reconciliation, as Lincoln didn't believe in the South being treated as a vanquished foe to be ruled by a victorious North.