"South Carolina was the first to leave the Union and form a new nation called the Confederate States of America. Four months later, six other states seceded. They were Georgia, Florida, Alabama, Mississippi, Texas and Louisiana. Later Virginia, Arkansas, North Carolina, and Tennessee joined them."
That's how we're told it happened. But how? Was it a governor's action? State legislature? Were the votes unanimous? Did the votes or acts have language to them?
Secession was initiated by the state governments on a sort of civic level, usually involving state assemblies and via resolutions passed through the appointed delegates. In each instance, the governor or other high-ranking political figures (i.e., Senators or House Rep's) led the charge, and had a heavy hand in drumming up support to push the assemblies and delegates towards the desired separation. As you said, South Carolina was the first to secede, and followed this route. The "fire-eaters" of the south were looking for a spark that could set in motion a formal meeting of state representatives from slave states to gather/organize in earnest about the logistics of mass secession, and South Carolina's resolution proved to be just that. Understand, this was an important consideration, as a state that seceded didn't stand much of a chance at staying that way unless they all banded together.
The Montgomery Convention in Feb. 1861 allowed various states that would largely form the basis of the future Confederacy to organize a provisional government and draft a Constitution. From this foundation the C.S.A. was brought to life, and allowed other states to secede INTO something rather than just away from the U.S.
[Sources, Bruce Catton, 'Mr. Lincoln's Army'; Shelby Foote, 'The Civil War: A Narrative, vol. 1'; James McPherson, 'Battle Cry of Freedom']