The political goal of the Union was to preserve the Union. President Lincoln took the position from the beginning that the Confederate States did not have the right to leave the Union, and in fact never had; rather they were simply "in rebellion. In fact, few people realize that in the annals of U.S. History, the official name of the Civil War is "The War of the Rebellion."
The primary goal of the Confederacy was to gain recognition by France and more importantly, Great Britain. Britain purchased a tremendous amount of southern cotton the proceeds of which funded the war for the South. At such time as they gained recognition from Britain, they could expect more aid from Britain; there was even the possibility that Britain might come into the war on the side of the South to protect its economic interests. This idea was not only practical, it was also a concern for Lincoln. Aside from all the high sounding oratory about the Emancipation Proclamation, Lincoln's primary purpose was to prevent Britain from supporting the South. Britain had outlawed slavery, and by issuing the Proclamation--which only freed slaves in states in rebellion, Lincoln forced Britain into a moral dilemma.
The most important political goals of the two sides were related to one another. The Confederacy was hoping to get recognized by European countries (particularly England and France) while the Union wanted to prevent this from happening.
For the Confederacy, recognition by an important country would have been a huge step towards winning their independence. The Union would have been hard pressed to continue fighting for the return of territory that had actually become an independent country in the eyes of the international community. Because of this, the Confederacy tried hard to get recognized and the Union tried to prevent it. These were the major goals that were specifically political.